Use the change-of-base formula for logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or e, we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.

To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.

Given any positive real numbers M, b, and n, where n1 and b1, we show

logbM=lognMlognb

Let y=logbM. By taking the log base n of both sides of the equation, we arrive at an exponential form, namely by=M. It follows that

{logn(by)=lognMApply the one-to-one property.ylognb=lognMApply the power rule for logarithms.y=lognMlognbIsolate y.logbM=lognMlognbSubstitute for y.

For example, to evaluate log536 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

{log536=log(36)log(5)Apply the change of base formula using base 10.2.2266 Use a calculator to evaluate to 4 decimal places.

A General Note: The Change-of-Base Formula

The change-of-base formula can be used to evaluate a logarithm with any base.

For any positive real numbers M, b, and n, where n1 and b1,

logbM=lognMlognb.

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

logbM=lnMlnb

and

logbM=logMlogb

How To: Given a logarithm with the form logbM, use the change-of-base formula to rewrite it as a quotient of logs with any positive base n, where n1.

  1. Determine the new base n, remembering that the common log, log(x), has base 10, and the natural log, ln(x), has base e.
  2. Rewrite the log as a quotient using the change-of-base formula
    • The numerator of the quotient will be a logarithm with base n and argument M.
    • The denominator of the quotient will be a logarithm with base n and argument b.

Example 13: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change log53 to a quotient of natural logarithms.

Solution

Because we will be expressing log53 as a quotient of natural logarithms, the new base, = e.

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

{logbM=lnMlnblog53=ln3ln5

Try It 13

Change log0.58 to a quotient of natural logarithms.

Solution

Q & A

Can we change common logarithms to natural logarithms?

Yes. Remember that log9 means log109. So, log9=ln9ln10.

Example 14: Using the Change-of-Base Formula with a Calculator

Evaluate log2(10) using the change-of-base formula with a calculator.

Solution

According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base e.

{log210=ln10ln2Apply the change of base formula using base e.3.3219Use a calculator to evaluate to 4 decimal places.

Try It 14

Evaluate log5(100) using the change-of-base formula.

Solution