Logarithms
Grade 11 · Math · Study Guide
Master logarithms — the inverse of exponentiation — and learn to apply log properties to simplify and solve equations.
A logarithm answers the question: to what power must we raise the base to get a given number? Written log_b(x) = y, it means b^y = x. The common log uses base 10 and the natural log uses base e ≈ 2.718.
The three core properties — product, quotient, and power rules — let you break complex logs into simpler pieces. These properties come directly from the corresponding exponent rules.
The change-of-base formula allows you to evaluate any logarithm on a standard calculator: log_b(x) = log(x) / log(b).
| Property | Rule |
|---|---|
| Product Rule | log_b(MN) = log_b(M) + log_b(N) |
| Quotient Rule | log_b(M/N) = log_b(M) − log_b(N) |
| Power Rule | log_b(M^p) = p · log_b(M) |
| Identity | log_b(b) = 1 |
| Zero Rule | log_b(1) = 0 |
| Type | Base | Notation |
|---|---|---|
| Common logarithm | 10 | log(x) or log₁₀(x) |
| Natural logarithm | e ≈ 2.718 | ln(x) |
Key terms
- Logarithm
- The exponent to which a base must be raised to produce a given number; log_b(x) = y means b^y = x.
- Natural logarithm
- A logarithm with base e (≈ 2.718), written ln(x).
- Change-of-base formula
- log_b(x) = log(x) / log(b), used to evaluate logs in any base using a calculator.
- Logarithmic equation
- An equation that contains a variable inside a logarithm, solved by converting to exponential form.
Practice questions
What is log₂(32)?
- 4
- 5
- 6
- 3
Show answer
2^5 = 32, so log₂(32) = 5.
Which property allows log₃(27x) to be written as log₃(27) + log₃(x)?
- Power Rule
- Quotient Rule
- Product Rule
- Change-of-Base
Show answer
The product rule states log_b(MN) = log_b(M) + log_b(N).
Solve for x: log₄(x) = 3.
Show answer
Converting to exponential form: 4^3 = 64, so x = 64.
Using the change-of-base formula, log₅(20) equals which of the following?
- log(5) / log(20)
- log(20) / log(5)
- ln(5) / ln(20)
- 20 / 5
Show answer
The change-of-base formula gives log₅(20) = log(20) / log(5).
Condense into a single logarithm: 2·log(x) + log(y) − log(z).
Show answer
Apply the power rule to get log(x²) + log(y) − log(z), then product/quotient rules to get log(x²y / z).
Fun facts
- The Richter scale for earthquakes is logarithmic — a magnitude 7 quake releases about 31 times more energy than a magnitude 6.
- Your ears perceive sound logarithmically too; decibels are a log scale, which is why doubling the sound power only adds about 3 dB.
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