Change of Base formula

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This is one of the most vital sections for logarithms. We cover primary and secondary properties of logs, which are pivotal in future math classes as these properties are often exploited in otherwise difficult mechanical situations.

You can watch a lecture on this section!

The previous examples work nicely as long as we have logs with matching bases. In the unfortunate (but likely) situation where the bases don’t match, we will need a way to manipulate the bases so that we can use our above properties. This leads us to the ‘change of base’ formula for logarithms.

Write the logarithm $\log _2 x^2$ as a log with a base of $5$ We want to rewrite the log above as $\log _5($ something $)$ . Perhaps the easiest way to do this involves using the exponential form of the log, that is, if $\log _2(x^2) = y$ , then $x^2 = 2^y$ . We know we want log base 5, so we can apply this to both sides; getting $\log _5(x^2) = \log _5(2^y) = y\log _5(2)$ . Remember that $y$ is actually our original expression, so if we divide both sides by $\log _5(2)$ to solve for $y$ we get; $\log _2(x^2) = y = \frac {\log _5 \left (x^2\right )}{\log _5(2)}$ Which means we have successfully written our original expression as an expression involving only $\log _5$ logs.

If we generalize the previous example, we can actually derive the fully generalized change of base formula. If you want to change the base from $b$ to the base $c$ , then you can use the following equation;

$\log _b(x) = \frac {\log _c(x)}{\log _c(b)}$