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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!
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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 as a log with a base of We want to rewrite the log above as
something. Perhaps the easiest way to do this involves using the exponential form of
the log, that is, if , then . We know we want log base 5, so we can apply this to
both sides; getting . Remember that is actually our original expression,
so if we divide both sides by to solve for we get; Which means we have
successfully written our original expression as an expression involving only
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 to the base , then you
can use the following equation;
Precalculus Algebra