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Solving Exponential Equations with the Different Bases - 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!

Solving an exponential equation where our bases can’t be rewritten as the same. For this particular problem behind me what we’re looking at is e to a power is equal to 18. e is a decimal 2.7 so on so forth so we obviously can’t write 18 and that as the same base. So what we need to do is somehow get out variable, our x down to the same level so we can manipulate it. In order to do that what we have to do is take the log of both sides.

For this particular example though we’re dealing with a base of e. So there’s a special log that we can use in order to help us eliminate some terms. If I take the natural log of both sides, remember natural log is log base e, what we end up with is we bring this exponent down and around, this .04x natural log of e is equal to natural log of 18. But what you need to remember is the natural log of e is actually log base of e which is just 1. So this disappears all together leaving us with .04 x is equal to natural log of 18, because this is equal to 1.

Now we just have a simple equation to solve, x has a coefficient so we divide by that. X is equal to natural log of 18 divided by .04. Whenever you can, you have a log base e, take the natural log of t, you lose the term all together, makes the solving significantly easier.

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