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What Are Logarithmic Equations?

Logarithmic equations are where x is part of what you take the logarithm of. To get x out of the logarithm, we need to use the logarithmic rules or exponents to completely get rid of the logarithm. When you use log ⁡ , you use 10 as your base, when you use ln ⁡ you use e as your base.

Theory

Logarithmic equations

In logarithmic equations you get rid of the logarithm by taking the exponent on both sides either with base-10 or with base-e. You use base-10 when you are dealing with a Briggs logarithm and base-e when you are dealing with the natural logarithm.

log ⁡ x = c ln ⁡ x = c 10log ⁡ x = 10c eln ⁡ x = ec x = 10c x = ec

Note! You can only take the logarithm of positive numbers.

Example 1

Solve the logarithmic equation ln ⁡ x = 4

ln ⁡ x = 4 eln ⁡ x = e4 x = e4 x ≈54.6

Example 2

Solve the logarithmic equation 1 4 log ⁡ 2x = 8

1 4 ln ⁡ 2x = 8 | ⋅ 4 ln ⁡ 2x = 32 eln ⁡ 2x = e32 2x = e32 | ÷ 2 x = e32 2 ≈3.95 ⋅ 1013

Example 3

Solve the logarithmic equation 5 ln ⁡ (x + 2) = 15

5 ln ⁡ (x + 2) = 15| ÷ 5 ln ⁡ (x + 2) = 3 eln ⁡ (x+2) = e3 x + 2 = e3 x = e3 − 2 e3 − 2 ≈ 20.1 − 2 x ≈ 18.1

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