Definition

We define the logarithm to base a to be the inverse of a^x:

log_a(x) = (e^xlna)^-1 implies log_a(a^x) = x

Graph

Properties

1. log_a(1) = 0, log_a(a) = 1

2. log_a(uv) = log_a(u) + log_a(v)

3. log_a(u^v) = v log_a(u)

4.

Proof of 2

Proof of 3

Proof of 4

Derivative

Proof

Change of base

log_bu = (log_ba)( log_au)

Thus