Log (logarithm)

What is log

The log function isolates the exponent. It is the inverse function to exponentiation.

lo g_{a} (b) = lo g_{a} (a^{c}) = c

Example: 8 is the power of 2, $lo g_{2} (8) = lo g_{2} (2^{3}) = 3$

Note that $lo g_{2} (0)$ = no value, because $2^{x} = 0$ does not exist, even when $2^{- 1000} = \frac{1}{2 ^{1000}}$ is tiny, it’s not zero.

In R, $lo g_{2} (0) = - \infty$

After each experiment, the measurement will double, therefore log function is useful.

It scale the axis of fold changes (quantity changes) to a symmetric scale, so it’s important to use a log scale when describing fold changes.

Multiplication can be converted to logs.
$15 = 3 \times 5 = 2^{1.6} \times 2^{2.3} = 2^{1.6 + 2.3} = 2^{3.9}$
Multiplication within log can be converted to addition
$lo g_{a} (X \times Y) = lo g_{a} (X) + lo g_{a} (Y)$
Example:
$l o g_{2} (8) = lo g_{2} (2 \times 4) = lo g_{2} (2) + lo g_{2} (4) = 3$
Division converted to subtraction
$2/4 = 2^{1} \times 2^{- 2} = 2^{- 1}$
Division within log can be converted to subtraction
$lo g_{a} (\frac{X}{Y}) = lo g_{a} (X) - lo g_{a} (Y)$
Example:
$l o g_{2} (\frac{2}{4}) = lo g_{2} (2) - lo g_{2} (4) = - 1$