Concept 8

Work with exponential expressions/equations and logarithms when presented.

Rewrite matching bases first, and read logarithms as exponent questions.

Core Idea

For exponential equations, try rewriting both sides with the same base first. If a logarithm appears, treat it as another way to express an exponent.

Understanding

Rule: ACT exponential questions are usually about structure, not advanced log rules. If the bases can be rewritten to match, the exponents can be compared directly.

When a logarithm appears, read it as an exponent question. For example, log𝑏⁡𝑎 =𝑐 means 𝑏𝑐 =𝑎. Switching between those two forms is often all you need.

Step by Step

  1. Rewrite both sides with the same base when possible.
  2. Compare exponents once the bases match.
  3. If a logarithm appears, convert it to exponent form.

Misconceptions

  • Trying to compare exponents before the bases match.
  • Treating logarithms as unrelated notation instead of another way to write exponents.
  • Missing the simpler rewrite that makes the equation easy.
Question

Worked Example

If 9𝑥 =27, what is the value of 𝑥?

Select an answer to see the explanation