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?