Rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. log5 x

Analyzing functions
asked 2020-11-20
Rewrite the logarithm as a ratio of
(a) common logarithms and
(b) natural logarithms. \(\displaystyle{\log{{5}}}{x}\)

Answers (1)

To evaluate logarithms to any base, we use the change-of-base formula: \(\displaystyle{\log{{c}}}{a}=\frac{{{\log{{b}}}{a}}}{{{\log{{b}}}{c}}}\)
a) \(\displaystyle{\log{{5}}}{x}=\frac{{\log{{x}}}}{{\log{{5}}}}\)
This is a common logarithm as the base is changed from 5 to 10 the same way the logarithm is changed from base “c” to base “b” in the formula.
b) \(\displaystyle{\log{{5}}}{x}=\frac{{\ln{{x}}}}{{\ln{{5}}}}\)
To change \(\displaystyle{\log{{5}}}\) x to ratio of a natural logarithm, change \(\displaystyle{\log{{5}}}\), also written as ln.

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