How to Solve Logarithms With Different Bases

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A logarithmic expression in mathematics takes the form

y = \log_bx

where ​y​ is an exponent, ​b​ is called the base and ​x​ is the number that results from raising the ​b​ to the power of ​y​. An equivalent expression is:

b^y = x

In other words, the first expression translates to, in plain English, "​y​ is the exponent to which ​b​ must be raised to get ​x​." For example,

3 = \log_{10}1,000

because 103 = 1,000.

Solving problems that involve logarithms is straightforward when the base of the logarithm is either 10 (as above) or the natural logarithm ​e​, as these can easily be handled by most calculators. Sometimes, however, you may need to solve logarithms with different bases. This is where the change of base formula comes in handy:

\log_bx = \frac{\log_ ax}{\log_ab}

This formula allows you to take advantage of the essential properties of logarithms by recasting any problem in a form that is more easily solved.

Say you are presented with the problem

y = \log_250

Because 2 is an unwieldy base to work with, the solution is not easily imagined. To solve this type of problem:

Step 1: Change the Base to 10

Using the change of base formula, you have

\log_250 = \frac{\log_{10}50}{\log_{10}2}

This can be written as log 50/log 2, since by convention an omitted base implies a base of 10.

Step 2: Solve for the Numerator and Denominator

Since your calculator is equipped to solve base-10 logarithms explicitly, you can quickly find that log 50 = 1.699 and log 2 = 0.3010.

Step 3: Divide to Get the Solution

\frac{1.699}{0.3010} = 5.644


If you prefer, you can change the base to ​e​ instead of 10, or in fact to any number, as long as the base is the same in the numerator and the denominator.



About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at