# Solving Exponent Problems Similarly, if is a positive integer, we define to be . Otherwise we'd be dividing by .) How could we make sense of an expression like ?If you don't already know the answer, this is a good exercise; I recommend puzzling over it for awhile. The backwards (technically, the "inverse") of exponentials are logarithms, so I'll need to undo the exponent by taking the log of both sides of the equation.

Similarly, if is a positive integer, we define to be . Otherwise we'd be dividing by .) How could we make sense of an expression like ?If you don't already know the answer, this is a good exercise; I recommend puzzling over it for awhile. The backwards (technically, the "inverse") of exponentials are logarithms, so I'll need to undo the exponent by taking the log of both sides of the equation.

However, this value, while "exact", won't be very helpful for word problems (or in "real life") if you need a numerical approximation.

But we can't evaluate this expression in our calculators as it stands.

We want to do the opposite of multiplication four times. Therefore, It is also possible to extend the exponential function to all non-integers. Well, hoping that property 1 will remain true when , we see that should (hopefully) be equal to .

Listed below are some important properties of exponents: If is a number and each of and is a positive integer, then, as explained above (property 1), . For that reason, we define , in order to make that be true.

Hoping that property 1 will remain true even if or is negative, we see that should (hopefully) be equal to .

Thus, we define to be , in order to make this be true.

The Hydra was a one-headed monster but when it is cut off, 2 more heads grow in its place. It might not be actually easier to find the answer (without a calculator), but isn’t is neat to be able to use an exponent to write such a long multiplication problem in such a simple way?

If a hero tried to conquer it by cutting off all of its heads every day, how many heads would the Hydra have on the third day?

Similarly, the exponentiation is defined as the repetition of multiplication.

For example, writing out can get boring fast, so we define the exponential function to express this in a much more compact form so that the preceeding example can be written as (read 3 to the 5th or 3 to the 5 power).

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