Logarithms
Grade 10 · Math · Study Guide
Understanding logarithms as inverses of exponential functions and applying logarithm properties.
A logarithm answers the question: 'What exponent do I need to raise the base to, to get this number?' Formally, log_b(x) = y means bʸ = x. For example, log₂(8) = 3 because 2³ = 8. The three parts of a logarithm are the base, the argument (the number inside), and the result (the exponent).
Two special logarithms appear constantly. The common logarithm (log, base 10) is written simply as log(x). The natural logarithm (ln, base e ≈ 2.718) is written ln(x). These are the two buttons on your calculator. For any other base, you use the change of base formula: log_b(x) = log(x)/log(b) = ln(x)/ln(b).
The properties of logarithms let you expand or condense complex log expressions. The product rule: log(ab) = log(a) + log(b). The quotient rule: log(a/b) = log(a) - log(b). The power rule: log(aⁿ) = n·log(a). These mirror the exponent laws — multiplication becomes addition, division becomes subtraction, and exponents become multiplication. This is why logarithms were invented in the 17th century: before calculators, they converted multiplication into addition, making complex calculations far faster.
Solving logarithmic equations often involves converting to exponential form or using properties to combine terms. Always check solutions — the argument of a logarithm must be strictly positive, so solutions that make the argument ≤ 0 are extraneous.
| Property | Rule | Example |
|---|---|---|
| Product Rule | log(ab) = log a + log b | log(6) = log(2) + log(3) |
| Quotient Rule | log(a/b) = log a - log b | log(5/2) = log 5 - log 2 |
| Power Rule | log(aⁿ) = n·log a | log(8) = 3·log(2) |
| Change of Base | log_b(x) = log(x)/log(b) | log₅(25) = log(25)/log(5) = 2 |
Key terms
- Logarithm
- log_b(x) = y means bʸ = x; the exponent needed to raise b to get x.
- Common Log
- log(x) = log₁₀(x); base 10 logarithm.
- Natural Log
- ln(x) = log_e(x); base e logarithm, used extensively in science and calculus.
- Change of Base Formula
- log_b(x) = log(x)/log(b); converts any log to base 10 or base e for calculation.
- Power Rule of Logs
- log_b(xⁿ) = n · log_b(x); brings exponents in front as coefficients.
Flashcards
- log_b(x) = y means...
- bʸ = x (logarithm converts to exponential form)
- log₁₀(1000)
- 3, because 10³ = 1000
- Product rule
- log(ab) = log(a) + log(b)
- Quotient rule
- log(a/b) = log(a) - log(b)
- Power rule
- log(aⁿ) = n · log(a)
- Change of base formula
- log_b(x) = log(x) / log(b)
Practice questions
What is log₂(32)?
- 4
- 5
- 6
- 16
Show answer
Answer: 5
log₂(32) = y means 2ʸ = 32. Since 2⁵ = 32, y = 5.
log(a) + log(b) = log(___) by the product rule.
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Answer: ab
Product rule: log(a) + log(b) = log(a·b) = log(ab).
ln(x) uses base e, while log(x) uses base 10.
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Answer: True
ln is the natural logarithm with base e ≈ 2.718. log (written without a base) is the common logarithm with base 10.
Using the power rule, log(x³) equals:
- 3 + log(x)
- log(x) + log(3)
- 3 · log(x)
- log(3x)
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Answer: 3 · log(x)
Power rule: log(x³) = 3·log(x). The exponent moves to become a coefficient in front of the log.
log(-5) is a real number.
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Answer: False
The domain of logarithms is strictly positive real numbers. log(negative number) is undefined in the real number system.
Fun facts
- The Richter scale for earthquakes is logarithmic (base 10): a magnitude 7 earthquake releases about 31.6 times MORE energy than a magnitude 6 — not just one more unit.
- John Napier invented logarithms in 1614 specifically to make multiplication easier for astronomers — before calculators, multiplying large numbers was incredibly tedious.
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