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Solving Exponential Equations with the Different Bases - 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!

Solving a exponential equation when we can’t rewrite our bases as the same base. What we have here is 5 to a power is equal to 13. The first thing we have to, first you noticed that 5 and 13 can’t be written as the same base. What we first have to do is figure out a way to bring down this t. We’re solving for t so we need to somehow get it down to the level we know how to manipulate it. In order to do that we have to take the log of both sides. It doesn’t’ matter what base you’re using as long as the base is the same.

So for this example I’m just going to use the common log, log base 10, so we take the log of both sides. Once we take the log of both sides we now have an exponent which we can bring down in front using our power rule. What we end up with is –t plus 1 log of 5 equals log of 13. Now we just want to solve for t.

Log of 5 and log of 13 are ugly numbers, we don’t know what they are but they’re still just numbers, so we can deal with them just as we would anything else. We want to solve for –t. There’s number of different ways of dong that, we could distribute the log 5 into here, we could divide by log 5, doesn’t really matter, just a different couple of different options we're going to do.

I am going to divide by log 5. What we end up with then is -t plus 1 is equal to log 13 over log 5. Subtract 1, -t is equal to log 13 over log 5 minus 1 and then in order to get the t by itself we need to divide by -1, so we make the logs negative and the 1 positive.

Finishing up we end up with t is equal to negative log 13 over log 5 and then divide by -1, so that just turns into a positive 1. Taking the log of both sides in order to solve an exponential equation.

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