Logarithm laws

Logarithms follow exponent laws

Because logarithms are exponents, the exponent laws become logarithm laws.

Plus useful facts:

• log_b 1 = 0
• log_b b = 1
• log_b(bˣ) = x
• b^(log_b x) = x

Examples of each law

• log₂ 8 + log₂ 4 = log₂ 32 = 5
• log 500 − log 5 = log 100 = 2
• log₃ 9⁴ = 4 log₃ 9 = 4 × 2 = 8

Expanding

To expand, break a single log into parts.

log(x²y/z) = 2 log x + log y − log z

log₅(25√x) = log₅ 25 + ½ log₅ x = 2 + ½ log₅ x

Condensing

To condense, combine into one log (useful for solving equations).

3 log x + log y − 2 log z = log(x³y/z²)

log₂ 12 − log₂ 3 = log₂ 4 = 2

Change of base

log_b x = log x / log b (or ln x / ln b)

log₄ 50 = log 50 / log 4 ≈ 1.699/0.602 ≈ 2.82

What the laws do NOT say

• log(x + y) ≠ log x + log y
• log x × log y ≠ log(xy)
• (log x)² ≠ 2 log x
• log x / log y ≠ log(x/y)

Using the laws to solve exponential equations

5ˣ = 40 → x log 5 = log 40 → x = log 40 / log 5 ≈ 2.29

Worked examples

Example 1: Evaluate log₆ 4 + log₆ 9 = log₆ 36 = 2.

Example 2: Expand log₃(81x⁵) = 4 + 5 log₃ x.

Example 3: Condense ½ log a − 3 log b = log(√a / b³).

Common mistakes

• Splitting log(x + y).
• Forgetting the power law applies only to the power of the argument: log(2x)³ = 3 log 2x, not 3 log 2 + x.
• Combining logs with different bases.

Try it yourself

1. log 4 + log 25
2. log₂ 80 − log₂ 5
3. Expand log(a³/b).
4. Condense 2 log x + log 3.
5. Solve 2ˣ = 15 (two decimals).

Answers: 1) 2 2) 4 3) 3 log a − log b 4) log(3x²) 5) ≈ 3.91

Quick recap

• Product law: add; quotient law: subtract; power law: bring the exponent down.
• Expand or condense as needed.
• Change of base lets a calculator handle any base.