A logarithmic expression in mathematics takes the form

y = log_{b}x

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:

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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 10^{3} = 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_{b}x = log _{a}x/log _{a}b

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_{2}50. 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_{2}50 = 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

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.