Study guides › Level 8 · Algebra 2

Logarithms (logs)

Level 8 · Algebra 2 · for questions like “log₂ 32 = ?” or “log₃ 54 − log₃ 2 = ?”

💡 The idea

A log answers one question: the base to what power makes this number?

log₂ 8 = 3, because 2³ = 8.

The small number (here 2) is the base. The answer is a power.

🪜 The steps

1. Write the number as a power of the base. 81 = 3⁴, so log₃ 81 = 4.
2. Two logs with the same base? Combine them into one first:

- adding logs → multiply inside: log x + log y = log (x × y)

- taking away logs → divide inside: log x − log y = log (x ÷ y)

3. “One over” means a negative power: 1/25 = 5⁻², so log₅ (1/25) = −2.

Two to remember: log of the base itself is 1 (log₇ 7 = 1), and log of 1 is 0.

✏️ Worked examples

1. log₃ 81 = ?

3 × 3 × 3 × 3 = 81, so 81 = 3⁴. The answer is 4.

2. log₂ 4 + log₂ 8 = ?

Adding logs multiplies inside: log₂ (4 × 8) = log₂ 32. And 32 = 2⁵, so the answer is 5.

3. log₅ (1/25) = ?

25 = 5², so 1/25 = 5⁻². The answer is −2.

⚠️ Watch out

• A log is not a division. log₂ 8 is 3, not 8 ÷ 2 = 4.
• log x + log y is not log (x + y). Adding logs multiplies what's inside.
• The answer can be negative (for numbers smaller than 1). The number inside a log can't be.

🎯 Try these

1. log₁₀ 1,000 = ?
2. log₄ 64 = ?
3. log₃ 54 − log₃ 2 = ?
4. log₂ (1/16) = ?

Answers: 1) 3 2) 3 3) 3 (54 ÷ 2 = 27 = 3³) 4) −4

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.