# Logarithm Formula

Logarithms are reverse operations of exponents. Suppose that $a^n = b$ then $^a log b = n$ and vice versa (if $^a log b =n$ then $a^n = b$). Therefore,

$^a log b = n Leftrightarrow a^n = b$

with a logarithm principal number, $a > 0$, $a neq 1$, b the number that the logarithm looks for, $b > 0$ and n is the result of the logarithm (exponent).
To be able to work on logarithmic problems, use the following logarithmic properties.

1. $^a log b^n = n ^a log b$
2. $^a log (bc) = ^a log b + ^a log c$
3. $^a log ( frac{b}{c} ) = ^a log b – ^a log c$
4. $^a log b times ^b log c = ^a log c$
5. $^{a^n} log b^m = frac{m}{n} ^a log b$
6. $^a log b = frac{1}{^b log a}$
7. $a^{^a log b} = b$
8. $^a log b = frac{log b}{log a}$

Note: If the principal number of a logarithm is not written, then the mean number of the logarithm is 10. So $^{10} log 7$ is written with $log 7$ only.

Problems example:
1. If $^3 log 4 = p$ and $^2 log 5 = q$ then the value for $^3 log 5$ is …
2. Know $^2 log 5 = p$ and $^5 log 3 = q$. The value of $^3 log 10$ is expressed in p and q is …
3. Results of $^{ frac{1}{5}} log 625+ ^{64} log frac{1}{16} + 4 ^{(3 ^{25} log 5)}$ is …

begin{align} & ^2 log 5 = q \ & Leftrightarrow ^4 log 5^2 = q \ & Leftrightarrow 2 ^4 log 5 = q \ & Leftrightarrow ^4 log 5 = frac{q}{2} end{align}
begin{align} ^3 log 5 & = ^3 log 4 ( ^4 log 5 ) \ & = p frac{q}{2} \ & = frac{pq}{2} end{align}
begin{align} ^3 log 10 & = frac{log 10}{log 3} \ & = frac{^5 log 10}{^5 log 3} \ & = frac{^5 log (2 times 5)}{^5 log 3} \ & = frac{^5 log 2 + ^5 log 5} {^5 log 3} \ & = frac{frac{1}{p} + 1}{1} \ & = frac{1 + p}{pq} end{align}.