Change of Base Formula for Logarithms

Recall that a logarithm is an exponent.
For example, $\,\log_2 8\,$ (log base two of eight) is the power that $\,2\,$ must be raised to, to get $\,8\,$.

In this case, the numbers work out nicely: $\log_2 8 = 3\,$, since $\,2^3 = 8\,$.

But what if, say, you need to know $\,\log_2 9\,$?
You know it will be a little more than $\,3\,$, but suppose you need a six decimal place approximation?

Most calculators have only two built-in logarithms:

the natural logarithm (log base $\,\text{e}\,$), denoted by ‘$\,\ln\,$’
the common logarithm (log base $\,10\,$), often denoted by ‘$\,\log\,$’

You can rummage around your calculator menus looking for logarithms to bases other than $\,\text{e}\,$ or $\,10\,$,
but you're not likely to find them. What's a person to do?

The good news is that it is very easy to rename a logarithm as an expression involving a different base.
All that is needed is the Change of Base Formula for Logarithms, which is the subject of this section.

Here's a preview of coming attractions: $$ \begin{gather} \cssId{s18}{\text{changing to natural logs:}}\cr\cr \cssId{s19}{\log_2 9} \ \ \cssId{s20}{=\ \ \frac{\ln 9}{\ln 2}} \cssId{s21}{\overset{\text{calculator}}{\ \ \ \ \ \ \strut\approx\ \ \ \ \ \ } 3.169925}\cr\cr\cr \cssId{s22}{\text{changing to common logs:}}\cr\cr \cssId{s23}{\log_2 9} \ \ \cssId{s24}{=\ \ \frac{\log 9}{\log 2}} \cssId{s25}{\overset{\text{calculator}}{\ \ \ \ \ \ \strut\approx\ \ \ \ \ \ } 3.169925} \end{gather} $$

Indeed, you can change to any allowable base: e.g., $$\cssId{s27}{\log_2 9 = \frac{\log_7\, 9}{\log_7\, 2}}$$ However, this isn't a useful name for calculator computation.

You probably already see the pattern from these three examples.
Here's the precise statement:

Change of Base Formula for Logarithms

Let $\,a\,$ and $\,b\,$ be positive numbers that are not equal to $\,1\,$, and let $\,x\gt 0\,$.
Then, $$ \cssId{s35}{\log_b\,x} \cssId{s36}{=\frac{\log_a\,x}{\log_a\,b}} $$ In words:
You can change from any base $\,b\,$ to any base $\,a\,$;
the ‘adjustment’ is that you must divide by the log to the new base ($\,a\,$) of the old base ($\,b\,$).

Derivation of the Change of Base Formula for Logarithms

The following equations are equivalent:

$y=\log_b\,x\,$	Give a name, $\,y\,$, to the left-hand side of the Change of Base formula.
$b^y=x\,$	Write the equivalent exponential form of the equation.
$\log_a\, b^y = \log_a\,x$	Apply the function $\,\log_a\,$ to both sides of the equation. (For more advanced readers: equivalence comes from the fact that $\,\log_a\,$ is a one-to-one function.)
$y\ \log_a\, b= \log_a\,x$	Use a property of logs to bring the $\,y\,$ down.
$\displaystyle y = \frac{\log_a\,x}{\log_a\,b}$	Divide both sides by $\,\log_a\,b\,$. Compare with the first step!

WolframAlpha has no trouble with logarithms, no matter what the base is.
For example, try each of these:
log base 2 of 9
common log of 100
natural log of e^2

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Introduction to Logarithmic Functions

(MAX is 15; there are 15 different problem types.)