1.8 The Logarithm

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

If we raise a number $b>0$ ($b\ne 1$) 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 $$\begin{array}{}\text{(a)}& u=b^r\end{array}$$ then$$\begin{array}{}\text{(b)}& r={\log }_{b}u\end{array}$$

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

$$2^3=8\text{and}{10}^{-4}=0.0001$$

we have

$${\log }_{2}8=3\text{and}{\log }_{10}0.0001=-4.$$

• Because $b$ is positive, $b^r>0$ for any real number $r$. Thus if $u<0$, the expression ${\log }_{b}u$ will be meaningless.