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.

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.