How do you change log base without a calculator?

Category: Logarithms »

 
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Overview: Log Base Change Formula

This question has its own dedicated lesson ». Make sure to check it out if you need a more thorough read, or want to become an expert on how change of base shows up on tests.Now more than ever, we're encouraged to use a graphing calculator for logarithm work. Since the early 2010's in fact, software updates to the TI-84 make it even easier to work with logarithms that are not base $10$ or $e$, while before we had to rely on changing the base ourselves by hand to a familiar base like $e$ or $10$ that our calculator had buttons for.In the right setting, though, there is still a need to know how to convert a logarithm to an equivalent expression that uses a different base. Namely, this is useful when working on a more complicated algebra problem for which decimal approximations are useless or not appropriate (your calculator only gives decimal approximations, after all).If you have the expression$$\log_4 (x)$$and you want to change it to log base $8$, we follow the change of base formula. To change from base $b$ to base $c$, use the formula$$\boxed{\log_b (x) = \frac{\log_c (x)}{\log_c (b)}}$$For our example, the formula tells us that$$\log_4 (x) = \frac{\log_8 (x)}{\log_8 (4)}$$Teachers commonly ask for this in situations where these base numbers are related, since you can do all the work without a calculator. In this example, we would probably be expected to know that $\log_8 (4)$ is $2/3$, so we would finalize this problem as$$\log_4 (x) = \frac{\log_8 (x)}{2/3}$$$$=\frac{3 \log_8 (x)}{2}$$