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Change of Base Formula - Problem 1

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!

The change of base formula for logarithms is an easy to us formula which allows us to evaluate logs other than base 10 or base e. So what it allows us to do is take a log of any base and convert it into a quotient where the bases can be whatever we want them to be. Typically we choose base 10 or base e because that is what our calculator is but we could choose any base that we wanted to.

For this particular example we’re going to use the change of base formula in a slightly different way. Normally when we have logarithms we try to figure out what each term is. Try to figure out what log base 3 of 64 is. And we do that by saying what power of 3 will give us 64? The problem is that we don’t know anything. The same thing with the log base 3 of 8 we don’t know what power of 3 is, sorry we don’t know what power, 3 to what power gives us 8, get that clarified. What we can do is actually change our base formula backwards. So instead of going from a single logarithm to 2, we can go to 2 back to 1.

Our logs are the same base, which tells that we are in this form and so all we need to do is write it as a single log. The log of the denominator, the 8 comes up to the base and the log of the numerator stays as it is. What we’ve done is we’ve taken this quotient and rewritten it using the change of base formula as a single log. We now know what this is, log base 8 of 64, is saying 8 to what power is 64, that’s just going to be equal to 2. The change of base formula doesn’t always have to be taken a single log and writing it as 2 we could also use it going to the other direction taking two logarithms of the same base and condensing it down to one.

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