Logarithms are one of the key differentiators between the SAT and ACT. They appear on the ACT but NOT on the SAT, making this a must-study topic for ACT test-takers.
What Is a Logarithm?
A logarithm answers: "What power do I raise the base to?"
Think of it as the inverse of exponentiation:
- means
- means
The Three Properties You Must Know
Product Rule:
Quotient Rule:
Power Rule:
These three rules handle 90% of ACT log questions.
Special Values to Memorize
- for any base (because )
- for any base (because )
- (natural log)
Solving Log Equations
Step 1: Isolate the log expression.
Step 2: Convert to exponential form.
Step 3: Solve the resulting equation.
Example: Solve
Convert:
Solve: , so
Common Mistakes to Avoid
- : The product rule applies to , NOT .
- Forgetting domain restrictions: is only defined for . Always check your solution.
- Confusing and : usually means on the ACT. means .
ACT Pro Tip
On the ACT, most log questions fall into two categories: (1) evaluate a simple log by converting to exponential form, or (2) simplify using one property. You rarely need to chain multiple properties. If the question looks complicated, you are probably overthinking it.
Master logarithms with our Logarithms lessons — ACT-exclusive content with practice exercises.