How To Use Logarithms

Now that you know what logs are I’ll talk about how to use them. There are a few simple rules of logarithms that you’ll use to answer almost any log question. Each log rule is equivalent to a power rule, but if you get confused it might help to remember why we use logs: to turn multiplication into addition.

The Rules of Logs

If you remember your power rules, when you multiply the numbers you add the powers. So logs is the opposite: you multiply numbers by adding logs:

logbx + logby = logb(xy)

You can check this by example:

log24 = 2 (because 22 = 4)
log28 = 3 (because 23 = 8)
log24 + log28 = 2 + 3 = 5
log2(4 x 8) = log232 = 5 (because 25 = 32)

Note that – like all log rules – this only works if the base (b) is the same for both logs.

Here are the other log rules. I won’t give examples, you can easily make up your own if you want to try.

logbx - logby = logb(x/y)
a.logbx = logb(xa)

There are two special logs. In the same way that x0 = 1 and x1 = x we have:

logb1 = 0
logbb = 1

And finally, remember that log is the inverse of power. You can think of log and power as cancelling each other out. So:

logb(bx) = x
b(logbx) = x

Finally, not really a rule but a useful trick for turning numbers into logs:

x = x.1 = x.logbb = logbbx

For example:

6 = 6logbb = logbb6

With those few rules you should be able to answer any log question. In the next post I’ll show you my method for solving log questions.