How To Solve Logarithms With Different Bases

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

\(\frac{1.699}{0.3010} = 5.644\)

Note

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.