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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