Use the change-of-base theorem to find an approximation to four d...

Question

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_√19 5

Accepted Answer

Hello, everyone. We are asked to evaluate the following expression and approximate the answer to three significant figures using the change of base theorem. The expression we are given is log base radical 23 of eight. Our answer choices are a negative 1. B 0.512, C 1.33 D 0.663, rewriting our original logarithm again, we had the log and the base is radical of eight. Recall that the change of base theorem states that when we have a logarithm and in this case, we'd say base A of B, we can rewrite this and find a value by using a different base. Meaning I could do log base C of B divided by the log base C of A. In this case, the other base I'm gonna use is E so I can use the natural log. So log base radical 23 of eight using the change of base theorem can be looked at as the natural log of eight divided by the natural log of radical 23. I will be using a calculator to find the decimals. The natural lug of eight is approximately 2.794415. It doesn't end there, but I felt like that was enough digits. And then the natural log of radical 23 is approximately 1.567747. Now, when I divide those two numbers, I get 1.326388, we are asked to approximate this to three significant digits. So we have a one, a three, a two, we're gonna round to the 100th place. So with three significant figures, the value of this using the change of base theorem is 1.33 or one in 33 hundreds which is answer choice C have a nice day.

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Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_√19 5