ACT Math: Logarithms

Logarithms may look familiar depending on what math you’re taking (or have taken) in school. If they don’t look familiar, chances are they will scare you when you take the ACT Math test. Luckily, with a little bit of studying the mystery surrounding them will clear right up. And also, there usually aren’t more than one or two logarithms on the actual ACT. So let’s get started.
A logarithm is, by definition, a number to a given base is the exponent to which the base must be raised in order to produce that number.
If you are confused by this, don’t worry. This is pretty much useless information, unless you plan on majoring in mathematics in college. What is important is looking at these examples, and being able to see that this is just another way of writing out exponents.
Here is a great example of a logarithm:
log₂8 = 3
We know we’re dealing with exponents here, so we see that the first number 2, is our base. And we want it to equal 8. So in order to do that, we raise our base to the third power.
2³ = 8
Here are a few more examples.
log₄16 = 2 (4² = 16)
log 100 = 2 (10² = 100)
For the last one, when you see a “log” by itself, it means that the base number is 10. log = log(10).
This is simple enough, and hopefully it clears up most of the questions. We don’t usually go too in depth with logarithms, because the ACT is really only testing that you know the basics.
Like exponents, logarithms also have certain rules attached to them.
log (ab) = log(a) + log(b)
log (a/b) = log(a) – log(b)
log (a^x) = x log(a)
Memorizing these will definitely let you answer any logarithm question on the ACT correctly.
Logarithms are a lot less complicated than they look! Logarithms are essentially the inverse of exponents. We’re used to seeing exponents in a format like y = x^a. In “logs” that equation is equal to log_x(y) = a. Let’s look at an example with actual numbers:
3² = 9   is the equivalent of    log₃(9) = 2
We would read the logarithm out loud as “log-base 3 of 9 equals 2.” A helpful way to remember this is to notice that whatever is on the other side of the equals sign is the exponent, and that the tiny number is the exponent base.
 

Let’s take a short quiz to practice moving back and forth between exponents and logarithms (answers at the end)! Translate the exponents to logs and the logs to exponents:
1. 10³ = 1,000
2. Log₇(49) = 2
3. 5² = 25
Now solve a few logarithms for the missing info:
4. Log₃(27) = ?
5. Log_x(81) = 2
6. Log₂(x) = 5