ACT Math Exponential and Logarithmic Equations: Solve Using the Inverse Relationship

Exponential form: b^x=y. Logarithmic form: log_b(y)=x. They're equivalent. Example: 2^3=8 means log_2(8)=3. Converting between forms solves equations. To solve 2^x=16, convert to logarithmic: log_2(16)=x, so x=4 (since 2^4=16). To solve log_3(x)=2, convert to exponential: 3^2=x, so x=9. Choose the form that makes the equation solvable. If x is in an exponent, use logarithms. If x is in the argument of a log, use exponentials.

Example: Solve 5^(x+1)=125. Convert: log_5(125)=x+1. Since 5^3=125, log_5(125)=3. So x+1=3, x=2.

Mistake 1: Forgetting which form to use. If x is in the exponent, take the logarithm of both sides. If x is the argument of a log, exponentiate both sides. Mistake 2: Changing the base when converting. If it's 2^x, use log base 2, not natural log (unless specified). Mistake 3: Forgetting to isolate the exponential or log before converting. Solve 2(3^x)=18. Divide by 2 first: 3^x=9. Then convert: log_3(9)=x, so x=2. Simplify before converting forms.

During practice, write out which form you're using and show the conversion step explicitly. This habit prevents mistakes.

Problem 1: 3^x=27. Convert to log: log_3(27)=x. Since 3^3=27, x=3. Problem 2: log_2(x)=5. Convert to exponential: 2^5=x, so x=32. Problem 3: 2(5^(x-1))=50. Divide by 2: 5^(x-1)=25. Convert: log_5(25)=x-1. Since 5^2=25, x-1=2, x=3. Problem 4: log(x)+2=3. Subtract 2: log(x)=1. This means 10^1=x, so x=10. Problem 5: 4^(2x)=64. Convert: log_4(64)=2x. Since 4^3=64, log_4(64)=3. So 2x=3, x=1.5. Solve all five, showing form conversion and solving steps.

Find five exponential-log problems from a practice test. For each, identify which form to use, convert, and solve. By the fifth problem, equation solving will feel natural.

Exponential and log equation questions appear on some ACT Math tests, usually in questions 50-60. They test understanding of inverse relationships. Students who convert between forms fluently pick up 1 point because the inverse relationship makes solving mechanical.

Drill exponential-log conversions daily this week. Each day, solve five equations by converting between forms. By test day, you should solve any exponential or log equation within 90 seconds.