Logarithms are not incredibly common on the ACT, but you are likely to see 1, or maybe 2, amongst the harder problems on the ACT Math test, so if you are shooting for a top score on the ACT Math, they are worth knowing.
First of all,
What is a logarithm?
A logarithm is the power to which a number must be raised in order to produce some other number.
If you went, “Huh?” I don’t blame you.
But probably you know about exponents:
If we see 32 we know this is 3 x 3, which equals 9.
If we see 45 we know this is 4 x 4 x 4 x 4 x 4, which equals 1024.
Logarithms are about thinking about exponents in a different way.
Here’s the question logarithms answer: What is the power something is raised to in order to get a number that we know?
So let’s say we are trying to figure out what power 3 must be raised to to get 9. Well, we just figured that out above: it’s 2. And what if we are trying to figure out what power 4 must be raised to in order to get 1024? Again, we saw this above, it’s 5.
This brings us to the mathematical definition of a logarithm:
Definition of a logarithm
If logab = c, then ac = b
So you need to remember, “What power do I need to raise a to to get b?
Using our example above, what does log41024 = ______?
Hopefully you said 5.
Now try this one:
What is log264?
(think to yourself what power do I need to raise 2 to to get 64?)
The answer is 6. 26 = 64.
Change of Base Rule
If you have a scientific or graphing calculator, your calculator has a log button. But this log button only calculates bases of ten (log10). So one more important trick to remember about logs that will help you quickly convert logs of any base to ones you can plug in your calculator is the change of base rule. Here is is:
Logab = log a / log b (a base of 10 is implied when it is not written).
So if you see log48, you can convert this to log 8 / log 4 and plug this in your calculator to get the answer 3/2. Which if you check it back in the problem, it makes sense: 43/2 = 8.