ACT Math Logarithms: Master This Algebra Tool Without Memorizing Tables

Relationship 1: log_b(x)=y means b^y=x. Understanding this equivalence unlocks everything. Example: log_2(8)=3 means 2^3=8. Relationship 2: log(a×b)=log(a)+log(b) (product rule). Example: log(10×100)=log(10)+log(100)=1+2=3. Relationship 3: log(a/b)=log(a)-log(b) (quotient rule). Example: log(1000/10)=log(1000)-log(10)=3-1=2. These three relationships are all you need to solve ACT logarithm questions. The test doesn't require you to calculate logarithm values or use tables; it tests understanding of these three rules.

Quick drill: If log_3(x)=2, what is x? Use Relationship 1: 3^2=x, so x=9. What is log(100)+log(10)? Use Relationship 2: log(100×10)=log(1000)=3. Each takes 20 seconds once the relationships are clear.

Mistake 1: Forgetting the relationship between logarithms and exponents. If log_b(x)=y, then b^y=x. Not remembering this relationship prevents you from solving problems. Mistake 2: Confusing the properties. log(a+b) does NOT equal log(a)+log(b). Only log(a×b) breaks into a sum. Mistake 3: Assuming you need to calculate the actual logarithm value. ACT almost never asks you to compute log(345) without a calculator. Instead, questions use properties to simplify. Focus on properties and relationships, not calculation.

Create a card showing the three relationships and three examples. Reference it daily this week. By test day, these relationships will be automatic.

Problem 1: If log_5(x)=2, find x. Use Relationship 1: 5^2=x, so x=25. Problem 2: Simplify log(1000)+log(100). Use Relationship 2: log(1000×100)=log(100000)=5. Problem 3: Simplify log(x^3). Use power rule: 3log(x). Problem 4: If log_2(x)=5, find x. Use Relationship 1: 2^5=x, so x=32. Problem 5: Simplify log(10)-log(10). Use Relationship 3: log(10/10)=log(1)=0. Solve all five, identifying which relationship you used for each.

Find five logarithm questions from a practice test. For each, identify which property applies and solve using the three relationships. By the fifth question, logarithm solving will feel straightforward.

Logarithm questions appear on some (not all) ACT Math tests, usually in questions 45-60. They're harder than algebra but rewarding because they require only three relationships plus logical application. Students who master the three logarithm relationships pick up 1-2 points on the math section because logarithms intimidate students who haven't studied them, giving you an advantage.

Learn the three relationships this week and drill them daily. By test day, you should recognize any logarithm question and apply the appropriate relationship within 60 seconds.