Change of Base Formula Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Evaluate

\log_5(20)

using the change of base formula and a calculator.

1
The change of base formula: $\log_b(a) = \frac{\ln(a)}{\ln(b)}$ (or equivalently $\frac{\log(a)}{\log(b)}$ ).
2
Apply: $\log_5(20) = \frac{\ln(20)}{\ln(5)}$ .
3
Calculate: $\frac{\ln(20)}{\ln(5)} = \frac{2.9957}{1.6094} \approx 1.861$ .

Answer

\log_5(20) \approx 1.861

The change of base formula converts a logarithm from any base to a quotient of logarithms in a base your calculator supports (usually base 10 or base

e

). This is essential because most calculators only have

\log

and

\ln

buttons.

About Change of Base Formula

A formula for converting a logarithm from one base to another: $\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$ .

Show that [formula] using the change of base formula.

Simplify [formula] to an exact fraction.

Prove that [formula] using the change of base formula.