Logs

**johett** · January 15th 2008, 11:35 AM

simplify using logarithm properties
a)4log(subscript25)15-4log(25)3

Express the following as a single logarithm
c)1/3(log(5)X+log(5)Y)-4log(5)Z

Simplify
d)log(2)20-log(2)5

thank you

**wingless** · January 15th 2008, 12:09 PM

A summary of the basic properties:

1- $\text{log}_{c}a + \text{log}_{c}b = \text{log}_{c}ab$

2- $\text{log}_{c}a - \text{log}_{c}b = \text{log}_{c}(\frac{a}{b})$

3- $log_{a}{b} = \frac{\text{log}_{c}{b}}{\text{log}_{c}{a}}$ (c can be anything you want, don't forget that $c > 0$ and $c \neq 1$ )

4- $a \text{log}_{b}c = \text{log}_{b}(c^a)$

5- $\frac{1}{a}\text{log}_{b}c = \text{log}_{(b^a)}c$

Last two properties combined:
6- $\frac{c}{d}\text{log}_{a}{b} = \text{log}_{(a^d)}{(b^c)}$

7-Let's say that $x = \text{log}_{a}{b}$
If I multiply it by $\frac{c}{c}$
$x = \text{log}_{a}{b} = \frac{c}{c}\text{log}_{a}{b}$
From property 6,
$x = \text{log}_{(a^c)}{(b^c)}$
Which means,
$\text{log}_{a}{b} = \text{log}_{(a^2)}{(b^2)} = \text{log}_{(a^3)}{(b^3)} = \text{log}_{(a^4)}{(b^4)} = \text{log}_{(a^{1/2})}{(b^{1/2})}....$

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A)
$4\text{log}_{25}{15} - 4\text{log}_{25}{3}$

$4(\text{log}_{25}{15} - \text{log}_{25}{3})$

$4\text{log}_{25}{\frac{15}{3}}$ ...(From property 2)

$4\text{log}_{25}{5}$

$4\frac{1}{2} = \boxed{2}$

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B)
$\frac{1}{3}(\text{log}_{5}{x} + \text{log}_{5}{y}) - 4 \text{log}_{5}{z}$

$\text{log}_{(5^3)}{x} + \text{log}_{(5^3)}{y} - 4 \text{log}_{5}{z}$

$\text{log}_{(125)}{x} + \text{log}_{(125)}{y} - 4 \text{log}_{125}{z^3}$

$\text{log}_{(125)}{xy} - 4 \text{log}_{125}{z^3}$

$\text{log}_{(125)}{xy} - \text{log}_{125}{z^{12}}$

$\text{log}_{(125)}{\frac{xy}{z^{12}}}$

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C)
$\text{log}_{2}{20} - \text{log}_{2}{5}$

$\text{log}_{2}{4}$

$\boxed{2}$