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.

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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. \(\color{blue}{-3}\)

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