3.2: Logarithmic Notation

A Logarithm is an exponent. In the early $1600^{\prime}$ s, the Scottish mathematician John Napier devised a method of expressing numbers in terms of their powers of ten in order simplify calculation. since the advent of digital calculators, the methods of calculation using logarithms have become obsolete, however the concept of logarithms continues to be used in many area of mathematics.

The fundamental idea of logarithmic notation is that it is simply a restatement of an exponential relationship. The definition of a logarithm says:

$$\log _{b} N=x \rightarrow b^{x}=N$$
The notation above would be read as "log to the base $b$ of $N$ equals $x$ means that $b$ to the $x$ power equals $N . "$ In this section we will focus mainly on becoming familiar with this notation. In later sections, we will learn to use this process to solve equations.

Example
Express the given statement using exponential notation:

$$\log _{2} 32=5$$
If $\log _{2} 32=5,$ then $2^{5}=32$

Example
Express the given statement using exponential notation:
$\log _{7} 4 \approx 0.7124$
If $\log _{7} 4 \approx 0.7124,$ then $7^{0.7124} \approx 4$

If the logarithm notation appears without a base, it is usually assumed that the base should be 10

Example
Express the given statement using exponential notation:

$$\begin{array}{l} \log 100=2 \\ \text { If } \log 100=2, \text { then } 10^{2}=100 \end{array}$$

The notation $\ln N=x$ is typically used to indicate a logrithm to the base $e$. This means that:

$$\ln N=x \rightarrow e^{x}=N$$

Example
Express the given statement using exponential notation:
$\ln 15 \approx 2.708$
If $\ln 15 \approx 2.708,$ then $e^{2.708} \approx 15$
In some cases, we would want to change an exponential statement into a logarithmic statement.

Example
Express the given statement using logarithmic notation:
$12^{4}=20,736$
If $12^{4}=20,736$ then $\log _{12} 20,736=4$

Example
Express the given statement using logarithmic notation:
$10^{2.5} \approx 316.23$
If $10^{2.5} \approx 316.23$ then $\log 316.23 \approx 2.5$

Example
Express the given statement using logarithmic notation:
$e^{6} \approx 403.4$
If $e^{6} \approx 403.4,$ then $\ln 403.4 \approx 6$

Exercises 3.2
Rewrite each of the following using exponential notation.
1) $\quad t=\log _{5} 9$
2) $\quad h=\log _{7} 10$
3) $\quad \log _{5} 25=2$
4) $\quad \log _{6} 6=1$
5) $\quad \log 0.1=-1$
6) $\quad \log 0.01=-2$
7) $\quad \log 7 \approx 0.845$
8) $\quad \log 3 \approx 0.4771$
9) $\quad \log _{2} 35 \approx 5.13$
10) $\quad \log _{12} 50 \approx 1.5743$
11) $\quad \ln 0.25 \approx-1.3863$
12) $\quad \ln 0.989 \approx-0.0111$

Rewrite each of the following using logarithmic notation.
13) $\quad 10^{2}=100$
14) $\quad 10^{4}=10,000$
15) $\quad 4^{-5}=\frac{1}{1024}$
16) $\quad 5^{-3}=\frac{1}{125}$
17) $\quad 16^{\frac{3}{4}}=8$
18) $\quad 8^{\frac{1}{3}}=2$
19) $\quad 10^{1.3} \approx 20$
20) $\quad 10^{0.301}=2$
21) $\quad e^{3} \approx 20.0855$
22) $\quad e^{2} \approx 7.3891$
23) $\quad e^{-4} \approx 0.0183$
24) $\quad e^{-2} \approx 0.1353$