Solving Exponent Problems

Solving Exponent Problems-43
However, this value, while "exact", won't be very helpful for word problems (or in "real life") if you need a numerical approximation.

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

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Answer: Hoping that property 1 will remain true when or is a fraction, we see that should (hopefully) be equal to .

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

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.

This article is an introduction to what exponentiation is and how it works.

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. The abbreviation is pronounced "ell-enn" and written with a lower-case "L" followed by a lower-case "N". Since science uses the natural log so much, and since it is one of the two logs that calculators can evaluate, I tend to take the natural log of both sides when solving exponential equations.This is not (generally) required, but is often more useful than other options.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).As with most problems in basic algebra, solving large exponents requires factoring.If you factor the exponent down until all the factors are prime numbers – a process called prime factorization – you can then apply the power rule of exponents to solve the problem.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.

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