What is Log

1. First law If $log_{a}x=n$ than $a^{n}=x$
2. Second law $log_{n}A+log_{n}B=log_{(}AB)$
3. Thirde law $log_{A}-log_{B}=log_{(}{A \over B})$
4. law If "${log_{x}}^{log_{x}}={log_{x}}^{2}=n$" than "${log_{x}}={\sqrt[{2}]{n}}$"
5. law if $log_{(}x^{n})={log_{x}}^{n}=n*{log_{x}}$
6. law "$log_{n}(x)={log_{n}(a)}$" than "x = a"

examples

example 1

ex1: $log_{4}4=n$ find n.

$log_{a}x=n$ $a^{n}=x$

$4^{n}=4$

n=1

example 2

ex2: $log_{4}x=2$ find x.

$log_{a}x=n$

$a^{n}=x$

$4^{2}=x$

x = 16

example 3

ex3: $log_{a}25=2$ find a.

$log_{a}x=n$

$a^{n}=x$

$a^{2}=25$

a = 5

example 4

ex4: $log_{a}25=2$ find a.

$log_{a}x=n$

$a^{n}=x$

$a^{2}=25$

a = 5

example 5

ex5: $log_{4}x+log_{4}(2x+5)=2$ find x.

1. Second law $log_{A}+log_{B}=log_{(}AB)$.

$log_{4}x+log_{4}(2x+5)=log_{4}(2x^{2}+4x)$

$log_{4}(2x^{2}+4x)=2$

$4^{2}=(2x^{2}+4x)$

$2x^{2}+4x-16=0$

x= 2 and -4

example 6

ex6: $log_{4}x+log_{4}(x+4)-log_{4}(x^{4}+8x^{3}+16x^{2})=2$ simplify

$log_{4}x+log_{4}(x+4)-log_{4}(x^{4}+8x^{3}+16x^{2})=2$

$log_{4}x+log_{4}(x+4)=2+log_{4}(x^{4}+8x^{3}+16x^{2})$

1. Second law $log_{n}A+log_{n}B=log_{(}AB)$

$log_{4}x+log_{4}(x+4)=log_{4}(x^{2}+4x)$

$log_{4}(x^{2}+4x)-log_{4}(x^{4}+8x^{3}+16x^{2})=2$

1. Thirde law $log_{A}-log_{B}=log_{(}{A \over B})$

$log_{4}(x^{2}+4x)-log_{4}(x^{4}+8x^{3}+16x^{2})=log_{4}({x^{2}+4x \over x^{4}+8x^{3}+16x^{2}})=2$

$log_{4}({1 \over x^{2}+4x})=2$