I’ve talked about the basics of logarithms and log rules, so let’s move on to actually **solving log equations**.

As usual with maths, there are different ways of tackling the problem. This is my general procedure, use whatever method works best for you:

1) Rearrange the equation to get all the logs on one side and all the numbers on the other

2) Use log rules to simplify the log side to a single log

3) Do ‘base to the power’ on both sides to get rid of the log

Here’s an example question:

Solve: log_{3}x + 4 = log_{3}7 + 6

First rearrange to get the logs on on side and the numbers on the other:

log_{3}x - log_{3}7 = 6 - 4 = 2

Now use log rules on the left. We can do this because the base is the same (3) in both expressions.

log_{3}(x/7) = 2

Do “3 to the power” on both sides to cancel out the log:

x/7 = 3^{2}= 9 x = 63

That’s the basic method. Watch out for trick questions where the base is *not* the same:

Solve: log_{6}x = 1 + log_{5}25

Because the bases are different we *can’t* put the logs together. Instead we need to recognise that log_{5}25 = 2. So we have:

log_{6}x = 1 + 2 = 3 x = 6^{3}= 216