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Solving equations where the variable is in an exponent usually involves use of logarithms. But before we do that, it will make things simpler if we get rid of the 5 first. So we'll start by dividing both sides by 5:
Now we'll use logarithms. It doesn't really matter which base of logarithm we use but it makes things simpler if you use the same base for the logarithm as the base that has an exponent. So we will use base e (ln) logarithms:
Now we use a property of logarithms, , to move the exponent of the argument out in front of the logarithm. This gets the variable out of the exponent! In fact it is this property that is the reason we use logarithms on problems like this: to get the variable out of the exponent.
By definition, ln(e) = 1 so this simplifies to:
Now we have just one thing left. Divide both sides by 0.045:
This is the exact answer for x. If you want a decimal approximation, get out your calculator and calculate the expression on the left.
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