Properties of Logarithms and Exponents

Let's consider the general exponential functions -- y = a^x and y = log_ax -- where a is a positive constant. We want to define exactly what these exprssions mean and what operations we can perform on them.

For any integers n and m :

aⁿ = a*a*a*...*a (n factors). a⁰ = 1a¹ = a

a^-n^ &sp;=&sp; 1 a^n^

(a^m)(a^m) = a^n+m

a^n^a^m^&sp;=&sp; a^n-m^

>/tr> (aⁿ)^m = a^nm

We're also interested in fractional exponents. Then, any rational number n can be expressed as $p q$ for some integers p and q. Then, $a^1 q^$ is defined at the qth root of a. [Sorry -- we don't have a way to display the mathematical symbol for that right now.] So, $a^p q^ &sp;=&sp;
(a^1 q^) ^p^$ or in other words, is the qth root of a raised to the pth power. That takes care of integers and rational numbers, which leaves us with irrational numbers to deal with.

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jjnichol@mit.edu