How to Change Base Formula for Logarithms?

Changing the base formula, as the name suggests, is used to change the base of the logarithm. You may have noticed that a scientific calculator only has “\(log\)” and “\(ln\)” buttons.

We know that “\(log\) ” stands for a logarithm of base \(10\) and “\(ln\)” stands for a logarithm of base \(e\). But there is no option to calculate the logarithm of a number with bases other than \(10\) and \(e\). Changing the base formula solves this problem.

Related Topics

• Properties of Logarithms
• How to Solve Natural Logarithms Problems

A step-by-step guide to the change of base formula for logarithms

The change of base formula is used to write a logarithm of a number with a definite base as the ratio of two logarithms each with the same base that is different from the base of the original logarithm. This is the property of logarithms.

Change of base formula

The change of base formula is:

\(\color{blue}{log _b\left(a\right)=\frac{\left[log _c\left(a\right)\right]}{\left[log _c\left(b\right)\right]}}\)

In this formula:

• The logarithm argument in the numerator is the same as the original logarithm argument.
• The logarithm argument at the denominator is the same as the base of the original logarithm.
• The base of both numerator and denominator logarithms must be the same, and this base can be any positive number other than \(1\).

Note: Another form of this formula is, \(log _b\left(a\right). log _c\left(b\right)=log _c\left(a\right)\), which is also widely used in solving problems.

The Change of Base Formula for Logarithms – Example 1:

Evaluate the value of \(log _{64}\left(8\right)\) using the change of base formula.

Solution:

Apply the base formula change by changing the base to \(10\):

\(log _{64}\left(8\right)=\frac{log\:8}{log\:64}\)

\(=\frac{log\:8}{log\:8^2}\)

\(=\frac{log\:8}{2\:log\:8}\)

\(=\frac{1}{2}\)

Exercises for the Change of Base Formula for Logarithms

Evaluate each logarithm by using the change of base formula.

1. \(\color{blue}{log _{\frac{1}{4}}\left(32\sqrt{2}\right)}\)
2. \(\color{blue}{log _{16}\left(32\right)}\)
3. \(\color{blue}{log _{\frac{1}{27}}\left(243\right)}\)
4. \(\color{blue}{log _{0.1}\left(1000\right)}\)

1. \(\color{blue}{-\frac{11}{4}}\)
2. \(\color{blue}{\frac{5}{4}}\)
3. \(\color{blue}{-\frac{5}{3}}\)
4. \(\color{blue}{-3}\)