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Exponential and Logarithmic Functions

This page is meant to serve as a quick overview of exponential and logarithmic functions.

**Additional Properties of Exponents**

If *a* > 0 and *a* ≠ 1, then *a*^{x} is a unique real number for all real numbers *x*.

If *a* > 0 and *a* ≠ 1, then *a*^{b} = *a*^{c} if and only if *b* = *c*.

If *a* > 1 and *m* < *n*, then *a ^{m} * <

If 0 <

**Properties of Logarithms**

For any positive real nyumbers *x* and *y*, real number *r*, and positive real number *a*, a ≠ 1.

log_{a}*xy* = log_{a}*x* + log_{a}*y*

log_{a} (*x*/*y*) = log_{a}*x* - log_{a}*y*

log_{a}*x*^{r} = *r*log_{a}*x*

log_{a}*a* = 1

log_{a}1 = 0

**Change-of-Base Theorem**

For any positive real nyumbers *x*,*a*, and *b* where *a* ≠ 1 and *b* ≠ 1.

log_{a}*x* = log_{b}*x* / log_{b}*a*

**Property of Logarithms**

If *x* > 0, *y* > 0, *a* > 0, and *a* ≠ 1, then
*x* = *y* if and only if log_{a}*x* = log_{a}*y*