CHANGING THE BASE OF A LOGARITHM

Let a, b, and x be positive real numbers such that tex2html_wrap_inline118 and tex2html_wrap_inline120 (remember x must be greater than 0). Then tex2html_wrap_inline122 can be converted to the base b by the formula

displaymath124

Let's verify this with a few examples.

Example 1: Find tex2html_wrap_inline126 to an accuracy of six decimals. Note that the answer will be between 1 and 2 because tex2html_wrap_inline128 and tex2html_wrap_inline130 , and 7 is between 3 and 9. According to the change of logarithm rule, tex2html_wrap_inline126 can be written tex2html_wrap_inline134 .

When the base is 10, we can leave off the 10 in the notation. Therefore tex2html_wrap_inline134 can be written tex2html_wrap_inline138 . Using your calculator,

displaymath140

You will note that the answer is between 1 and 2.

Let's check the answer. If tex2html_wrap_inline142 , our answer is correct. tex2html_wrap_inline144 . Close enough. Why isn't it 7 exactly? Well we rounded tex2html_wrap_inline138 to six places, so our answer won't check exactly. If we rounded to ten places, then when we checked the answer, it would be closer to 7 than this answer.

Example 2: We could work the same problem by converting to the base e. According to the change of logarithm rule, tex2html_wrap_inline126 can be written tex2html_wrap_inline150 . When the base is e, we can leave off the e in the notation and tex2html_wrap_inline150 can be written tex2html_wrap_inline154 . Using your calculator,

displaymath156

You will note that the answer is between 1 and 2.

Example 3: Find tex2html_wrap_inline158 . We know that tex2html_wrap_inline160 and tex2html_wrap_inline162 , and that 18 is between 16 and 32; therefore, we know that the exponent we are looking for is between 4 and 5. In fact, it is closer to 4 than to 5 because 18 is closer to 16 than it is to 32.

Let's solve this problem by changing the base to 10. tex2html_wrap_inline158 can be written tex2html_wrap_inline166 . Using your calculator,

displaymath168

Let's check the answer. If tex2html_wrap_inline170 , our answer is correct. tex2html_wrap_inline172 . Close enough. Why isn't it 18 exactly? Since we rounded tex2html_wrap_inline166 to six places, our answer won't check exactly. If we rounded to ten places, then when we checked the answer, it would be closer to 18 than this answer.

Example 4: We could work the problem in Example 3 by converting to the base e. According to the logarithmic rule, tex2html_wrap_inline158 can be written tex2html_wrap_inline178 . Using your calculator,

displaymath180

The answer is the same as the answer you found when you converted the base to 10.

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

Work the following problems and click on answer if you want to check your answer and review the solution.

Problem 1: Find tex2html_wrap_inline184 .

Answer.

Problem 2: Find tex2html_wrap_inline186 .

Answer.

Problem 3: Find tex2html_wrap_inline188 .

Answer.

Problem 4: Convert tex2html_wrap_inline190 to the base 2.

Answer.

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Author: Nancy Marcus

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