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,

To be able to work on logarithmic problems, use the following logarithmic properties.

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} $.

$ ^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$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.

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} $

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__:
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