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Solving Exponential Equations with the Different Bases - Problem 3

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!

Solving an exponential equation where our bases can’t be rewritten as the same base. Behind me I have an example where we’re dealing with 10 to a power is equal to 8. What we need to do is solve for x.

First thing we have to do is somehow get this x down to a level we can manipulate it. In order to do that we have to use logarithms. But before that what we need to realize is that we’re dealing with a base 10. There’s a special base, log base 10, the common log that will allow us to get rid of a log base 10 of 10. We take the log of both sides and bring down our exponent, this is going to come around, what we end up with is x plus 1 times the log of 10 is equal to log of 8.

Now, I chose common log because this term is just going to be equal to 1. Log of 10 is 1, leaving us with x plus 1 is equal to log 8. In order to solve this I subtract 1, x is equal to log 8 minus 1, and we know how to plug in log 8 into our calculator because it’s a common log, log base 10.

Whenever you’re dealing with an exponential equation where you’re dealing with the base of 10, always take the log base 10 of both sides, terms disappear, it makes your life a little bit easier.

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