The University of New South Wales
School of Mathematics and Statistics
Mathematics Drop–in Centre
LOGARITHMS
“Logarithm” is really just another word for “exponent” or
“power”. In an expression abwe call athe base and bthe loga-
rithm. So if
ab=c
then we say that bis the logarithm of cto the base a, written
b= logac.
In all of these equations, aand cmust be positive numbers; b
may be positive, negative or zero. In principle, logarithms can be
evaluated by rewriting them as powers.
Example. Evaluate log232.
Solution. Let x= log232. Then 2x= 32 and so by trial and
error x= 5.
In practice, evaluating logarithms usually requires a calculator.
For instance, if x= log234 then 2x= 34; the previous example
shows that xmust be a bit more than 5 and definitely less than
6, but it’s difficult to pin it down much more closely than this.
Logarithm laws. The following identities hold, where a, x, y are
positive numbers and pis any real number:
loga(xy) = logax+ logay
logax
y= logax−logay(∗)
loga(xp) = plogax .
In these equations it does not matter what base we use for the
logarithms, so long as it is the same throughout the equation.
So we shall often write simply log(xy) = log x+ log yand so on.
Combining the above properties we have, for example,
7 log(x2y5)−9 log(xy4)
= 7(2 log x+ 5 log y)−9(log x+ 4 log y)
= 5 log x−log y
= logx5
y.
Note that there is no useful simplification for log(x±y).
Further identities follow from the definition of the logarithm:
alogax=xand loga(ax) = x .
In this case the base of the logarithm does matter and cannot be
omitted. A sample simplification:
100log10 x=10log10 x2
=x2.
An especially important logarithm is the one with base e=
2·71828 ···. It is called the natural logarithm and written ln.
That is, ln xmeans the same as logex. All of the identities (∗)
remain true when logais replaced by ln. For example,
lne3
√x= ln(e3)−ln(x1/2) = 3 ln e−1
2ln x= 3 −1
2ln x .
Comment. We have ignored certain difficulties in defining loga-
rithms, and a more careful approach will be taken in MATH1131
lectures. Nevertheless, the properties given will remain true and
we can still manipulate expressions as in the previous examples.
