# CHANGING THE BASE OF A LOGARITHM - Example

Let a, b, and x be positive real numbers such that neither a nor b equals 1. Then can be converted to the base b by the formula . Let's verify this with the following example.

Example 1: Find .

Solution: Convert to the base 10.

Check: Let's check the answer. If equals 125, we have worked the problem correctly. = 124.999443. Close enough. The reason the check does not come out exactly is that we rounded the log125, the , and the quotient of the two logs. If you use continuous calculations on your calculator, the answer will be closer to 125.

Solution 2: You can also work this problem using natural logarithms. Convert to the base e.

Check: Let's check the answer. If equals 125, we have worked the problem correctly. . Close enough. The reason the check does not come out exactly is that we rounded the ln125, the , and the quotient of the two logs. If you use continuous calculations on your calculator, the answer will be closer to 125.

If you would like to review another example of changing the base of a logarithm, click on Example.

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