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 n≠1 and b≠1, we show
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
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.
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 n≠1 and b≠1,
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.
and
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 n≠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.
- 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, n = 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.
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.
Candela Citations
- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175. License: CC BY: Attribution. License Terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.