# Logarithm(log, lg, ln)

If b = a^{c} <=> c = log_{a}b

a, b, c are *real numbers* and b > 0, a > 0, a ≠ 1

**a** is called **"base"** of the logarithm.

Example: 2^{3} = 8 => log_{2}8 = 3

the base is 2.

Logarithm - an animated example

There are standard notation of logarithms if the base is 10 or e.

_{10}b = lg b

log

_{e}b = ln b

#### Properties of logarithms:

log_{a}1 = 0

log_{a}a = 1

a^{logab} = b

log_{a}(b.c) = log_{a}b + log_{a}c

log_{a}(^{b}/_{c}) = log_{a}b - log_{a}c

log_{a}b^{n} = n.log_{a}b

$log_ba=\frac{1}{log_ab}$

$log_bc = \frac{log_ac}{log_ab}$

$log_{a^n}b = \frac{1}{n}log_ab, \ \ n\ne0$

log_{a}(b ± c) - there are no such a formula.

#### Antilogarithm

log_{a}b = log_{a}c ⇔ b = c

log_{a}b = c ⇔ a^{c} = b, where b > 0, a > 0 and a ≠ 1

log_{a}b > log_{a}c ⇔ if a > 1 then b > c,

if 0 < a < 1 then b < c

#### Logarithmic calculator

**log**

_{2}=

#### Graphs of logarithmic functions

It shows that when x = 1, log = 0; when x -> 0 => log -> -∞; when x -> ∞ log -> ∞

If you have any question go to our forum about logarithms.