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exponentiallogs

# exponentiallogs - In this section well take a look at...

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In this section we’ll take a look at solving equations with exponential functions or logarithms in them. We’ll start with equations that involve exponential functions. The main property that we’ll need for these equations is, Example 1 Solve . Solution The first step is to get the exponential all by itself on one side of the equation with a coefficient of one. Now, we need to get the z out of the exponent so we can solve for it. To do this we will use the property above. Since we have an e in the equation we’ll use the natural logarithm. First we take the logarithm of both sides and then use the property to simplify the equation. All we need to do now is solve this equation for z .

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Example 2 Solve . Solution Now, in this case it looks like the best logarithm to use is the common logarithm since left hand side has a base of 10. There’s no initial simplification to do, so just take the log of both sides and simplify. At this point, we’ve just got a quadratic that can be solved So, it looks like the solutions in this case are and . Now that we’ve seen a couple of equations where the variable only appears in the exponent we need to see an example with variables both in the exponent and out of it. Example 3 Solve . Solution The first step is to factor an x out of both terms. DO NOT DIVIDE AN x FROM BOTH TERMS!!!! Note that it is very tempting to “simplify” the equation by dividing an x out of both terms. However, if you do that you’ll miss a solution as we’ll see.
So, it’s now a little easier to deal with. From this we can see that we get one of two possibilities.

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