Changing base in logarithms

4 Changing base in logarithms

It is sometimes required to express the logarithm with respect to one base in terms of a logarithm with respect to another base.

Now

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

where we have used logs to base $a$ . What happens if, for some reason, we want to use another base, $p$ say? We take logs (to base $p$ ) of both sides of $b = a^{n}$ :

${log}_{p} (b) = {log}_{p} (a^{n}) = n {log}_{p} a (using one of the logarithm laws)$

$n = \frac{{log}_{p} (b)}{{log}_{p} (a)} that is {log}_{a} b = \frac{{log}_{p} (b)}{{log}_{p} (a)}$

This is the rule to be used when converting logarithms from one base to another.

Key Point 9

{log}_{a} b = \frac{{log}_{p} b}{{log}_{p} a}

For base 10 logs:

${log}_{a} b = \frac{log (b)}{log (a)}$

For example,

${log}_{3} 7 = \frac{log 7}{log 3} = \frac{0.8450980}{0.4771212} = 1.7712437$

(Check, on your calculator, that $3^{1.7712437} = 7$ ).

For natural logs:

${log}_{a} b = \frac{ln (b)}{ln (a)}$

For example,

${log}_{3} 7 = \frac{ln 7}{ln 3} = \frac{1.9459101}{1.0986123} = 1.7712437$

Of course, ${log}_{3} 7$ cannot be determined directly on your calculator since logs to base 3 are not available but it can be found using the above method.

Task!

Use your calculator to determine the value of ${log}_{21} 7$ using first base 10 then check using base $e$ .

Re-express ${log}_{21} 7$ using base 10 then base $e$ :

${log}_{21} 7 = \frac{log 7}{log 21} = 0.6391511$ ${log}_{21} 7 = \frac{ln 7}{ln 21} = 0.6391511$

Example 5

Simplify the expression $1 0^{log x}$ .

Solution

Let $y = 1 0^{log x}$ then take logs (to base 10) of both sides:

$log y = log (1 0^{log x}) = (log x) log 10$

where we have used: $log A^{k} = k log A$ . However, since we are using logs to base 10 then $log 10 = 1$ and so

$log y = log x implying y = x$

Therefore, finally we conclude that

$1 0^{log x} = x$

This is an important result true for logarithms of any base. It follows from the basic definition of the logarithm.

Key Point 10

a^{{log}_{a} x} = x

Raising to the power and taking logs are inverse operations.

Exercises

Find the values of
1. $log 8$
2. $log 50$
3. $ln 28$
Simplify
1. $log 1 - 3 log 2 + log 16.$
2. $10 log x - 2 log x^{2}$ .
3. $ln (8 x - 4) - ln (4 x - 2)$ .
4. $ln 10 log 7 - ln 7$ .

1. 3
2. 1.41096
3. 3.033
1. $log 2$ ,
2. $6 log x$ or $log x^{6}$ ,
3. $ln 2$ ,
4. $0$