ACT Prep Math Resource

The Ultimate ACT Guide to Logarithms

Logarithms (or "logs") can look intimidating because they use a brand new mathematical language. But here is the secret for the ACT: a logarithm is just an exponent running in reverse. The test usually features 1 to 2 logarithm questions. If you know how to convert them and apply three basic rules, you can solve almost every single one of them!

1The Core Definition: The "Loop" Trick

The most important skill for ACT logs is knowing how to rewrite a log equation as a regular exponent equation.

log b (x) = y \Leftrightarrow b y = x

Read this as: "Base b, raised to the power of y, equals x."

The "Loop" Method

To convert any log equation, start at the base, draw a loop to the other side of the equal sign (your exponent), and loop back to the inside number.

Example 1: Solve for x in

log 2 (8) = x

Rewrite it using the loop: $2 x = 8$
Since $23 = 8$ , we know that x = 3.

Example 2: Solve for x in

log x (9) = 2

Rewrite it using the loop: $x 2 = 9$
Since $32 = 9$ , we know that x = 3. (Note: Base numbers must always be positive).

2The Big Three Log Properties

The ACT loves to test your ability to expand or condense logarithms using three fundamental properties. Think of these as the cousins of the regular exponent rules.

Rule Name	The Property	How to think about it
Product Rule	$log b (M \cdot N) = log b (M) + log b (N)$	Multiplication inside becomes addition outside.
Camp/Quotient Rule	$log b (M / N) = log b (M) - log b (N)$	Division inside becomes subtraction outside.
Power Rule	$log b (M k) = k \cdot log b (M)$	An exponent inside can fly to the front as a multiplier.

ACT Application Example

Question: If $log a (x) = 3$ and $log a (y) = 5$ , what is $log a (xy 2)$ ?

Step 1: Use the Product Rule to separate x and y²:
$log a (x) + log a (y 2)$
Step 2: Use the Power Rule to bring the 2 to the front:
$log a (x) + 2 log a (y)$
Step 3: Substitute the given values:
$3 + 2(5) = 13$

3Two Special Logs to Look Out For

Sometimes the ACT will drop a log on you without a visible base number. Don't panic—these are just shortcuts!

The Common Log: If you see a log with no base written, the base is automatically 10.
$log(100) \to log 10 (100) = 2$ (because $102 = 100$ )
The Natural Log (ln): If you see "ln", it just means log_e, where e is a constant number approximately equal to 2.718. It follows all the exact same rules as normal logs.
$ln(e 5) \to log e (e 5) = 5$

Quick ACT Log Checklist

Stuck? Use the Loop Method to turn the log into a familiar exponent equation.
Are two logs with the same base being added? Multiply their insides together.
Are two logs with the same base being subtracted? Divide their insides.
Is there a coefficient in front of the log? Move it inside as an exponent.
See a plain log with no base? Write in a little 10 to keep your bearings.