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: log3x + 4 = log37 + 6
First rearrange to get the logs on on side and the numbers on the other:
log3x - log37 = 6 - 4 = 2
Now use log rules on the left. We can do this because the base is the same (3) in both expressions.
log3(x/7) = 2
Do “3 to the power” on both sides to cancel out the log:
x/7 = 32 = 9 x = 63
That’s the basic method. Watch out for trick questions where the base is not the same:
Solve: log6x = 1 + log525
Because the bases are different we can’t put the logs together. Instead we need to recognise that log525 = 2. So we have:
log6x = 1 + 2 = 3 x = 63 = 216
