ACT Math: Solve Logarithmic Equations Using Three Core Properties

Property 1 (Product Rule): log(a)+log(b)=log(a×b). Combine logs by multiplying the arguments. Property 2 (Quotient Rule): log(a)-log(b)=log(a/b). Subtract logs by dividing the arguments. Property 3 (Power Rule): n×log(a)=log(a^n). A coefficient in front of a log moves inside as an exponent. Example: Solve log(x)+log(2)=log(6). Use property 1: log(2x)=log(6). Drop the logs: 2x=6, so x=3. Example: Solve 2log(x)=log(16). Use property 3: log(x²)=log(16). Drop the logs: x²=16, so x=4 (or x=-4, but logs require positive arguments). Once you recognize which property applies, logarithmic equations become straightforward algebra.

Key insight: If logs with the same base appear on both sides, you can drop them and solve what remains. This converts a log equation into a simple algebraic equation.

Trap 1: Forgetting that log arguments must be positive. If your answer is x=-4 but the original equation has log(x), reject x=-4 because log of a negative number is undefined. Trap 2: Dropping logs incorrectly. You can only drop them if they have the same base on both sides. log(x)=2 cannot become x=2; it becomes x=10² (if base 10) or x=e² (if natural log). Trap 3: Mixing up log properties. log(a)+log(b) is NOT log(a+b); it is log(a×b). Trap 4: Using the wrong base. ACT usually uses base 10 (common log) or base e (natural log, written "ln"). If the base is not stated, assume base 10. Always check: Is the base clear? Are all arguments positive in my final answer?

Before you submit an answer, plug it back into the original equation to verify. This catches careless errors with domain restrictions and base confusion.

Equation 1: log(x)+log(3)=log(12). (Solution: log(3x)=log(12), so 3x=12, x=4.) Equation 2: 2log(x)=log(4). (Solution: log(x²)=log(4), so x²=4, x=2.) Equation 3: log(x)-log(2)=log(5). (Solution: log(x/2)=log(5), so x/2=5, x=10.) Equation 4: log(x)+log(x)=log(9). (Solution: log(x²)=log(9), so x²=9, x=3.) Equation 5: 3log(2)=log(x). (Solution: log(2³)=log(x), so log(8)=log(x), x=8.) For each equation, identify which property applies, rewrite using that property, then solve the resulting algebraic equation.

After solving, verify by substituting your answer back into the original equation. If the left side equals the right side, your solution is correct.

Logarithm problems appear 0-2 times per test and test your understanding of inverse operations (logs undo exponents) and algebraic manipulation. These are not trick questions; they are straightforward applications of the three properties. Once you own the properties and can match them to problem patterns, logarithm questions become reliable points.

This week, solve 10 logarithmic equations from old tests, using the three properties systematically. Time yourself; each should take 1-2 minutes. By test day, you will recognize logs instantly and solve them without hesitation, earning points that other students skip.