# Logarithm property

An important area of application for base 10 logarithms is when you want to solve equations containing x as an exponent.

Example
$6^{x}=20$
Now that we know that a number may be rewritten as an exponent of 10, we can start by rewriting 6 and 20:

$6=10^{\log 6}$

$20=10^{\log 20}$
We can insert these notations in an equation again:
$(10^{\log 6})^{x}=10^{\log 20}$
A rule from a earlier chapter:
$(a^{b})^{c}=a^{bc}$

Provides:
$10^{x\log 6}=10^{\log 20}$
Now we can see that both exponents equal each other and the left-hand side = right-hand side:
$x\cdot \log 6=\log 20$
Here we need only isolate x:
$x=\frac{log20}{log6}\approx 1.67$
There are some properties of logarithms that are important to master:

### First the product property

$log_{b}ac=log_{b}a+log_{b}c$

$a, b,c\; positive\; numbers,\; b\neq 1$

### Second the quotient property

$log_{b}\frac{a}{c}=log_{b}a-log_{b}c$

$a, b,c\; positive\; numbers,\; b\neq 1$

### Third the power property

$log_{b}a^{c}=c\cdot log_{b}a$

$c=real\; number,a\; and\; b\; positive\; numbers,\; b\neq 1$

## Video lesson

We wish to invest $1 000 in a fund that pays 12% in interest. How many years would it take for this investment to grow until we have$ 10 000 in the fund?