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Power Rule of Logarithms - Problem 2

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!

Another property of logarithms is that whenever we have a base to a power that’s a logarithm of the same base, those bases actually end up cancelling out and just leaving with whatever is inside the log of the numerator, of the exponent rather.

So putting this in the practice we have 5 log base 5 of 7, our bases are the same so these are just going to cancel out and leave us with 7. Easy enough, this one is a little bit different. We have a base of 6 to another to a log of base of 6 as well. But the difference here is we have this 2. In order for this theorem to hold true we can’t have anything else we just need that log. So what we have to do is use the power of logarithms just looking at the 2 log 6 of 3, we can bring this 2 up and around and end up with 6 log base 6 of 3². Just using the power of the other term up front you can bring it up or bring it back down.

So we end up with 6 log base 6 of 3² now we have our bases the same this is just going to end up being whatever is inside the log which is 3² or 9. This theorem it looks a little bit weird but it’s actually really straight forward, so what I want to do right now is just take one second and show you how it comes up.

So if we are dealing with log base b x is equal to y. That's just an equation you can take this equation and put it into exponential form which gives us b to the y is equal to x, and then just using a simple substitution. This whole thing is y, so plugging that back in we end up with b to the log base b of x is equal to x.

So it does look a little bit obstruct but really all we are doing is taking our formula putting it in our exponential form with a little substitution. So simplifying something whenever you have a base to a log if the bases are the same you answer is whatever is inside the log.

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