Definition
A logarithm answers the question "what exponent do I need?" Formally:
log_b(x) = y means bʸ = x
So log₃(81) = 4 because 3⁴ = 81, and log₁₀(0.01) = −2 because 10⁻² = 0.01. The base b must be positive and not 1, and the input x must be positive. The notation log without a base means base 10; ln means base e (the natural logarithm).
Logarithms undo exponentials
Because the two are inverses, log_b(bˣ) = x and b^(log_b x) = x. Special values: log_b(1) = 0 and log_b(b) = 1. Also ln(e⁴) = 4.
The three rules
- Product: log_b(MN) = log_b M + log_b N
- Quotient: log_b(M/N) = log_b M − log_b N
- Power: log_b(Mᵖ) = p · log_b M
Notice what is not true: log(M + N) is not log M + log N, and (log M)³ is not log(M³).
Change of base
To get a value on a calculator, use log_b(x) = ln x / ln b. For example log₅(20) = ln 20 / ln 5 ≈ 2.996/1.609 ≈ 1.861.
Worked example 1: condense
Simplify log₂(40) − log₂(5).
- Use the quotient rule: log₂(40/5) = log₂(8).
- Since 2³ = 8, the answer is 3.
Worked example 2: unknown exponent
Solve 3ˣ = 20.
- Take the log of both sides: ln(3ˣ) = ln 20.
- Power rule: x · ln 3 = ln 20.
- Divide: x = ln 20 / ln 3 ≈ 2.9957 / 1.0986 ≈ 2.727.
- Check: 3^2.727 ≈ 20. ✓.
Solving log equations
Rewrite in exponential form, then solve.
Solve log₂(x + 1) = 3. Then x + 1 = 2³ = 8, so x = 7. Check: log₂(8) = 3. ✓.
If an equation has logs on both sides with the same base, you can equate the insides, but always substitute back; the input to any log must stay positive.
Domain
The domain of log(x − 5) is x > 5, since x − 5 must be positive. The graph has a vertical asymptote at x = 5.
Natural exponent equations
To solve eˣ = 12, take ln of both sides: x = ln 12 ≈ 2.485.
Common mistakes
- Treating log(x + y) as a sum of logs.
- Forgetting to check that the answer makes every log argument positive.
- Mixing up which number is the base and which is the argument.