Exponents and logarithms are two sides of the same idea. Exponents ask "what do I get if I multiply this base by itself n times?" Logarithms ask the reverse: "how many times do I multiply this base to get a number?" Learn the exponent rules first, and the log rules will look familiar.
The exponent rules
- Product:
aᵐ · aⁿ = aᵐ⁺ⁿ. Example: x³ · x⁴ = x⁷. - Quotient:
aᵐ / aⁿ = aᵐ⁻ⁿ. Example: x⁸ / x³ = x⁵. - Power of a power:
(aᵐ)ⁿ = aᵐⁿ. Example: (x²)⁵ = x¹⁰. - Power of a product:
(ab)ⁿ = aⁿbⁿ. Example: (3x)² = 9x². - Zero exponent:
a⁰ = 1for any nonzero a. - Negative exponent:
a⁻ⁿ = 1 / aⁿ. Example: 2⁻³ = 1/8. - Fractional exponent:
a^(1/n)is the nth root of a, anda^(m/n)is the nth root of a raised to m. Example: 8^(2/3) = (cube root of 8)² = 2² = 4.
Worked example 1: simplify
Simplify (2x³y⁻²)² · x⁻⁴.
- Power of a product: 2²x⁶y⁻⁴ = 4x⁶y⁻⁴.
- Multiply by x⁻⁴: 4x²y⁻⁴.
- Rewrite without negative exponents: 4x² / y⁴.
What a logarithm means
log_b(x) = y means exactly the same thing as bʸ = x. So log₂(8) = 3 because 2³ = 8, and log₁₀(1000) = 3 because 10³ = 1000. When no base is written, "log" usually means base 10, and "ln" means the natural log with base e ≈ 2.718.
Two facts follow directly: log_b(1) = 0 (because b⁰ = 1) and log_b(b) = 1. You can only take the log of a positive number.
The log rules
- Product:
log_b(xy) = log_b(x) + log_b(y), matching the exponent product rule. - Quotient:
log_b(x/y) = log_b(x) − log_b(y). - Power:
log_b(xⁿ) = n · log_b(x). This is the rule that lets you solve for an exponent. - Change of base:
log_b(x) = log(x) / log(b), which lets you use a calculator for any base.
Worked example 2: expand a log
Expand log(x³ / y²).
log(x³) − log(y²) = 3 log(x) − 2 log(y).
Worked example 3: solve an exponential equation
Solve 5^x = 40.
- Take the log of both sides: log(5^x) = log(40).
- Power rule: x · log(5) = log(40).
- x = log(40) / log(5) ≈ 1.602 / 0.699 ≈ 2.29.
- Check: 5² = 25 and 5³ = 125, so an answer between 2 and 3 makes sense.
Worked example 4: exponential growth
A population of 2,000 bacteria doubles every 3 hours. When will it reach 50,000?
- Model: P = 2000 · 2^(t/3).
- 50,000 = 2000 · 2^(t/3), so 25 = 2^(t/3).
- Take logs: log 25 = (t/3) log 2, so t = 3 · log 25 / log 2 ≈ 3 × 4.64 ≈ 13.9 hours.
Common mistakes
- Adding exponents when you should multiply: (x²)³ is x⁶, not x⁵.
- Thinking a negative exponent makes the number negative. 2⁻² = ¼, which is positive.
- Splitting log(x + y) into log x + log y. There is no rule for the log of a sum.
- Writing (a + b)² as a² + b². Exponents do not distribute over addition.
Practice questions
- Simplify x⁵ · x⁻² / x².
- Evaluate 27^(2/3).
- Write log₃(81) = 4 as an exponential equation.
- Solve 2^x = 20 to two decimal places.
Answers: (1) x. (2) 9. (3) 3⁴ = 81. (4) x = log 20 / log 2 ≈ 4.32.
Last reviewed 2026-10-23 by the BrainBoost team.