# Logarithm(log, lg, ln)

If b = ac <=> c = logab
a, b, c
are real numbers and b > 0, a > 0, a ≠ 1
a
is called "base" of the logarithm.

Example: 23 = 8 => log28 = 3
the base is 2.

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

log10b is denoted by lg b
logeb is denoted by log b or ln b

#### List of logarithmic identities

loga1 = 0
logaa = 1
alogab = b

$\log_a(b \cdot c) = \log_ab + \log_ac$

$\log_a\frac{b}{c} = \log_ab - \log_ac$

$\log_ab^n = n \cdot \log_ab$

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

##### Changing the base

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

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

loga(b ± c) - there is no such a formula.

#### Antilogarithm

logab = logac ⇔ b = c
logab = c ⇔ ac = b, where b > 0, a > 0 and a ≠ 1

logab > logac ⇔ if a > 1 then b > c,
if 0 < a < 1 then b < c

Logarithm base:
log2 =

#### 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.

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