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 ...

Question Answer 1:
$ 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} $

Question Answer 2:
$ 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} $.

Question Answer 3: