>Logaritmelikninger>Hvordan løse logaritmeulikheter

Hvordan løse logaritmeulikheter

Når du løser logaritmeulikheter bruker du de vanlige reglene for ulikheter som du kjenner fra tidligere.

Regel

Logaritmeulikheter

Det å opphøye høyre og venstre siden i et grunntall gjør at ulikheten forblir uendret. Altså, ingenting skjer med ulikhetstegnet!

\begin{array}{l} \lg x & < c & \ln x & < c \\ 1 0^{\lg x} & < 1 0^{c} & e^{\ln x} & < e^{c} \\ x & < 1 0^{c} & x & < e^{c} \end{array}

Eksempel 1

Løs ulikheten $\ln (x - 4) \leq 8$

\begin{array}{l} lnEksempel 2 Løs ulikheten \lg (- 2 x + 3) \geq 4 \begin{array}{l} \lg (- 2 x + 3) & \geq 4 \\ e^{\ln (- 2 x + 3)} & \geq e^{4} \\ - 2 x + 3 & \geq e^{4} \\ - 2 x & \geq e^{4} - 3 & | : (- 2) \\ x & \leq \frac{e^{4} - 3}{- 2} = \frac{3 - e^{4}}{2} \end{array} Du snur ulikheten siden du dividerer med et negativt tall. Du må ha likhetstegn i siste rad  siden svaret kun skrives om slik at det ser penere ut!Eksempel 3 Løs ulikheten \ln (x - 3) \leq e \begin{array}{l} lnVil du vite mer? Registrer deg Det er gratis!Forrige oppslag Hvordan løse eksponentialulikheterTilbakeHjem All matte Spill Progresjon LigaOm oss Om House of Math Om ansatte Karriere Media Foredrag Blogg Kontakt support@houseofmath.com Booking av privatundervisning +47 22 150 300 Adresse Bygdøy allé 23
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Altså, ingenting skjer med ulikhetstegnet! \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\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmi  \u003ec\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                         \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  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class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eLøs\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eulikheten\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\u003cmi class=\"qopname\"\u003elg\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\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e≥\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003c/b\u003e\u003c/p\u003e                                                                                               \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003e  \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\u003cmi class=\"qopname\"\u003e lg\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmspace width=\"-0.17em\" 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class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\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≤\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e   \u003cmrow  \u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\u003e   \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mrow\u003e       \u003cmrow  \u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mrow\u003e\u003c/mfrac\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  Du snur ulikheten siden du dividerer med et negativt tall. Du må ha likhetstegn i siste rad  siden svaret kun skrives om slik at det ser penere ut!                                               \u003c/div\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                                               \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-1455\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Eksempel 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\"\u003eLøs ulikheten \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\u003cmi 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\u003e\u003cmi class=\"qopname\"\u003eln\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  \u003ex\u003c/mi\u003e\u003cmo  class=\"MathClass-bin\"\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e≤\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e        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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≤\u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ee\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-bin\"\u003e+\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e8\u003c/mn\u003e\u003cmi  \u003e,\u003c/mi\u003e\u003cmn\u003e1\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         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