# Solving Exponent Problems

Taking logarithms will allow us to take advantage of the log rule that says that powers inside a log can be moved out in front as multipliers.

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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.

To understand how exponents arise, let's first review how we can build multiplication from addition.

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. (And we only make this definition in the case where .We choose to leave undefined.) We can make sense of an expression like "" in a similar way.And, to solve an equation, I have to get the variable by itself on one side of the "equals" sign; to isolate the variable, I have to "undo" whatever has been done to the variable.In this case, the variable has been put in the exponent.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.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.This is useful to me because of the log rule that says that exponents inside a log can be turned into multipliers in front of the log: If you're asked to "find the solution", then the above should be an acceptable answer.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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