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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 ax is a unique real number for all real numbers x.
If a > 0 and a ≠ 1, then ab = ac if and only if b = c.
If a > 1 and m < n, then am < an.
If 0 < a < 1 and m < n, then am > an.

Properties of Logarithms

For any positive real nyumbers x and y, real number r, and positive real number a, a ≠ 1.
logaxy = logax + logay
loga (x/y) = logax - logay
logaxr = rlogax
logaa = 1
loga1 = 0

Change-of-Base Theorem

For any positive real nyumbers x,a, and b where a ≠ 1 and b ≠ 1.
logax = logbx / logba

Property of Logarithms

If x > 0, y > 0, a > 0, and a ≠ 1, then
x = y if and only if logax = logay