>Logarithmic Equations>How to Solve Exponential Inequalities

How to Solve Exponential Inequalities

When solving exponential inequalities it is very important that you look at the value of the argument $a$ in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by $\ln a$ or $\log a$ .

Rule

Solving exponential inequalities

When

a > 1

\ln (a) > 0

and you can solve the inequality as usual.

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & > \frac{\ln b}{\ln a} & x & > \frac{\log b}{\log a} \end{array}

When $0 < a < 1$ , $\ln (a) < 0$ , you will have to turn the inequality sign as you end up dividing or multiplying by a negative number!

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & < \frac{\ln b}{\ln a} & x & < \frac{\log b}{\log a} \end{array}

Example 1

Solve the inequality ${3.5}^{x} > 439$

\begin{array}{l} 3.5 x > 439 \ln {3.5}^{x} & > \ln 439 x \ln 3.5 & > \ln 439 & | \div \ln 3.5 x & > \frac{\ln 439}{\ln 3.5} \approx 4.9Example 2 Solve the inequality 50 \cdot {1.05}^{x} > 300 \begin{array}{l} 50 \cdot {1.05}^{x} > 300 | \div 50 {1.05}^{x} & > 6 \ln {1.05}^{x} & > \ln 6 x \ln 1.05 & > \ln 6 & | \div \ln 1.05 x & > \frac{\ln 6}{\ln 1.05} \approx 36.7Example 3 Solve the inequality 3 \cdot {0.25}^{x} > 27 \begin{array}{l} 3 \cdot {0.25}^{x} > 27 | \div 3 {0.25}^{x} & > 9 \lg {0.25}^{x} & > \lg 9 x \lg 0.25 & > \lg 9 & | \div \log 0.25 x & < \frac{\lg 9}{\log 0.25} \approx - 1.6Want to know more? Sign Up It's free!Previous entry What Are Logarithmic Equations? Next entry How to Solve Logarithmic InequalitiesGo backHome All Math Games Progress LeagueAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Booking of private tutoring +47 22 150 300 Address Bygdøy allé 23
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Discover the pitfalls that can crop up while working with these inequalities.","breadcrumbs":[{"path":"/encyclopedia","h1":"Encyclopedia","_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c"},{"path":"/encyclopedia/algebra","h1":"Algebra","_key":"e9327e4b-42fe-571b-bcb9-83fdb63d5f21"},{"path":"/encyclopedia/algebra/logarithms","h1":"Logarithms","_key":"5746e0ab-24ae-5841-af0f-f1b35cfab6c9"},{"path":"/encyclopedia/algebra/logarithms/logarithmic-equations","h1":"Logarithmic Equations","_key":"db5be093-ee70-5f75-bdd2-f3d4e77d9fda"},{"path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-exponential-inequalities","h1":"How to Solve Exponential Inequalities","_key":"22deaf56-6168-575e-8a39-4a3d360193ac"}],"key":"content/en/encyclopedia/02 Algebra/03 Logarithms/03 Logarithmic Equations/303ExponentialInequalities.tex","sharedLanguageKey":null,"path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-exponential-inequalities","otherLocales":[],"h1":"How to Solve Exponential Inequalities","siblings":[{"h1":"What Are Exponential Equations?","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-exponential-equations"},{"h1":"What Are Logarithmic Equations?","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/what-are-logarithmic-equations"},{"h1":"How to Solve Exponential Inequalities","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-exponential-inequalities"},{"h1":"How to Solve Logarithmic Inequalities","path":"/encyclopedia/algebra/logarithms/logarithmic-equations/how-to-solve-logarithmic-inequalities"}],"cssUrl":"/3e26151e-8cd9-466a-907f-43e1e928c4e8-en-0.css","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     When solving exponential inequalities it is very important that you look at the value of the argument \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e or \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-1460\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eSolving\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eexponential\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003einequalities\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eWhen \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/math\u003e, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e and you can solve the inequality as usual. \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                           \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace 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width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003eWhen \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003c/math\u003e, \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e, you will \u003cspan  class=\"ec-lmri-12x-x-144\"\u003ehave to \u003c/span\u003eturn the inequality sign as you end up dividing or multiplying by a negative number! \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                           \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                        \u003cmtd  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class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e                      \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                     \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e                                                                                              \u003c/div\u003e  \u003c/div\u003e                                                                                                                           \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-1461\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                            \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-label\"\u003e                            \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-label\"\u003e                            \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e                                   \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003c/b\u003e                                                                                                                     \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                               \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-1462\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                          \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                         \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                        \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-label\"\u003e                          \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                         \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e  \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003c/b\u003e                                                                                                                     \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                               \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-1463\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eSolve the inequality \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                            \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                  \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e÷\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003elg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" 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