Change of Base Formula for Logarithms

Before doing this exercise, you may want to review basic properties of 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:

Changing to natural logs: $$ \cssId{s19}{\log_2 9} \cssId{s20}{= \frac{\ln 9}{\ln 2}} \cssId{s21}{\strut\approx 3.169925} $$

Changing to common logs: $$ \cssId{s23}{\log_2 9} \cssId{s24}{= \frac{\log 9}{\log 2}} \cssId{s25}{\strut\approx 3.169925} $$

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