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Change of Base Formula - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When we encounter logarithms with bases not of the common or natural logarithm, we often need the change of base formula. The change of base formula allows us to convert a logarithm from one base to another. By using the change of base formula, we can change a logarithmic term to allow us to input it into a calculator. Most calculators only accept logarithms of base 10 or base e.

The change of base formula for logs, so if you look at your calculator what you notice is that you actually only have two buttons that relate to logarithms you have log and you have natural log okay? So those are the only two buttons that only two bases that your calculator can actually handle so if we want to tackle something like log base 2 of 7 we need to figure out a different way to enter this into our calculator okay? So by common sense, log base 2 of 7 is fairly close to log base 2 of 8 okay? So this is fairly close to log base 2 of 8, log base 2 of 8 is 3, 2 to the third is equal to 8 so I know that log base 7 is going to be a little bit less than 3 okay? But that typically is not going to really fly in terms of an answer you say oh it's going to be a little bit less than 3, so there has to be a way for us to figure out exactly what this is which is where the change of base formula comes in okay? So what that means, if we have log base b of x what this is equal to is log of x of any base over log of b of any base as long as those two bases are the same okay so sort of how I think about it is your b just drops down to become its own log, and then as long as these two boxes I have here are the same base this statement is equivalent to the statement over here okay, for most purposes because if we want to find an exact number we're going to have to put in a calculator. You're going to want to use either base 10 or base e the common or natural log okay? So let's go back to this log base 2 of 7 okay there's a couple ways we can rewrite this then okay? So log base 2 of 7 the one who used the common log, log base 10 we could say this is equal to log 7 over log 2 if we want to use the natural log what we can do is natural log of 7 over natural log of 2 or we could choose a base completely random let's say 84 and this would also be equal to the log base 84 of 7 over log base 84 of 2 okay? That last way is not very practical in terms of putting it into our calculator but just to show you how this all works it could be any base as long as they're the same.
Now point of clarification, this doesn't mean that the log of 7 is equal to the natural log of 7 okay? what it means is that the ratio of these two log of 7 over log of 2 is equal to the natural log of 7 over natural log of 2 okay, from here we could enter either of these to our calculator just by using the logarithm button, end up with the log of 7 divided by the log of 2 and this ends up to be 2.81 which like we said to begin with is going to be close to 3 but a little bit smaller. So the change of base formula it's really convenient for putting in different bases other than e or 10 into your calculator.

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