# 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}
So
\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}.