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 tex2html_wrap_inline78 can be converted to the base b by the formula tex2html_wrap_inline80 . Let's verify this with the following example.

Example 1: Find tex2html_wrap_inline82 .

Answer 1: -36.158608

Solution: Convert to the base 10.

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Check: Let's check the answer. If tex2html_wrap_inline88 equals 125, we have worked the problem correctly. tex2html_wrap_inline88 = 124.999443. Close enough. The reason the check does not come out exactly is that we rounded the log125, the tex2html_wrap_inline92 , 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 tex2html_wrap_inline82 to the base e.

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Check: Let's check the answer. If tex2html_wrap_inline98 equals 125, we have worked the problem correctly. tex2html_wrap_inline100 . Close enough. The reason the check does not come out exactly is that we rounded the ln125, the tex2html_wrap_inline102 , 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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Author: Nancy Marcus

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