1 Logarithms

Logarithms reverse the process of raising a base ‘ $a$ ’ to a power ‘ $n$ ’. As with all exponentials, the base should be a positive number.

If $b=a^n$ then we write ${log}_{a}b=n$ .

Of course, the reverse statement is equivalent

If ${log}_{a}b=n$ then $b=a^n$

The expression ${log}_{a}b=n$ is read

“The log to base $a$ of the number $b$ is equal to $n$ "

The term “log" is short for the word logarithm .

Example 3

Determine the log equivalents of

1. $16=2^4$ ,
2. $16=4^2$ ,
3. $1000=10^3$ ,
   
   
4. $134.896=10^{2.13}$ ,
5. $8.414867=e^{2.13}$

Solution

1. Since $16=2^4$ then ${log}_{2}16=4$
   
   
2. Since $16=4^2$ then ${log}_{4}16=2$
   
   
3. Since $1000=10^3$ then ${log}_{10}1000=3$
   
   
4. Since $134.896=10^{2.13}$ then ${log}_{10}134.896=2.13$
   
   
5. Since $8.41467={\text{e}}^{2.13}$ then ${log}_{\text{e}}8.414867=2.13$

Task!

Find the log equivalent of

1. $100=10^2$   
2. $\frac{1}{1000}=10^{-3}$

Here, on the right-hand sides, the base is 10 in each case so:

Answer

Task!

Find the log equivalent of

1. $b=a^n$ ,
2. $c=a^m$ ,
3. $bc=a^{n+m}$

1. Here the base is $a$ so:

   Answer

   $n={log}_{a}b$

2. Here the base is $a$ so:

   Answer

   $m={log}_{a}c$

3. Here the base is $a$ so:

   Answer

   $n+m={log}_a\left(bc\right)$

From the last Task we have found, using the property of indices, that

${log}_a\left(bc\right)=n+m={log}_{a}b+{log}_{a}c$ .

We conclude that the index law $a^n{a}^m=a^{n+m}$ has an equivalent logarithm law

${log}_a\left(bc\right)={log}_{a}b+{log}_{a}c$

In words: “The log of a product is the sum of logs."

Indeed this property is one of the major advantages of using logarithms. They transform a product of numbers (a relatively difficult operation) to a sum of numbers (a relatively easy operation).

Each index law has an equivalent logarithm law, true for any base, listed in the following Key Point: