In Section 3.2, we will study the logarithms in more detail.

If we raise a number $b>0$ ($b\neq1$) to a power $r$, compute the result, and obtain another number $u$, then $r$ is said to be the logarithm of $u$ to the base $b$ and we write $r=\log_{b}u$. That is, if $u=b^{r}\tag{a}$ then$r=\log_{b}u\tag{b}$

Formulas (a) and (b) are simply two different ways of expressing the same fact about the relation between $u$ and $r$. For example, because

$2^{3}=8\quad\text{and}\quad10^{-4}=0.0001$

we have

$\log_{2}8=3\quad\text{and}\quad\log_{10}0.0001=-4.$

• Because $b$ is positive, $b^{r}>0$ for any real number $r$. Thus if $u<0$, the expression $\log_{b}u$ will be meaningless.
The following properties of the logarithms immediately follow from Equations (a) and (b):

1.    $\log_{b}(uv)=\log_{b}u+\log_{b}v$

2.    $\log_{b}(u/v)=\log_{b}u-\log_{b}v$

3.    $\log_{b}(u^{n})=n\log_{b}u$

4.    $\log_{b}1=0$

5.    $\log_{b}(1/u)=-\log_{b}u$