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):
- \(\log_{b}(uv)=\log_{b}u+\log_{b}v\)
- \(\log_{b}(u/v)=\log_{b}u-\log_{b}v\)
- \(\log_{b}(u^{n})=n\log_{b}u\)
- \(\log_{b}1=0\)
- \(\log_{b}(1/u)=-\log_{b}u\)
We will study the logarithms in more detail in Section: Logarithmic Functions .