Logarithms Quiz

Grade 10 · Math · Quiz

Test your understanding of logarithm definitions, properties, and equations.

Questions

Evaluate log₃(81)
1. 3
2. 4
3. 9
4. 27
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Answer: 4
log₃(81) = y means 3ʸ = 81. Since 3⁴ = 81, y = 4.
log_b(b) = ___ for any valid base b.
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Answer: 1
log_b(b) = 1 because b¹ = b. Any base raised to the first power equals itself.
Expand: log(x²y/z) =
1. 2log(x)+log(y)-log(z)
2. 2log(x)·log(y)/log(z)
3. log(x²)+log(y)+log(z)
4. 2log(xy/z)
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Answer: 2log(x)+log(y)-log(z)
Apply rules: log(x²) + log(y) - log(z) = 2log(x) + log(y) - log(z). Power rule handles the x².
log_b(1) = 0 for any valid base b.
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Answer: True
log_b(1) = 0 because b⁰ = 1 for any base b. Any number raised to the 0 power is 1.
Solve: log₄(x) = 3
1. x = 12
2. x = 64
3. x = 7
4. x = 81
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Answer: x = 64
log₄(x) = 3 means 4³ = x. 4³ = 64, so x = 64.
Using change of base: log₅(100) = log(100) / log(___)
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Answer: 5
Change of base: log₅(100) = log(100)/log(5). The denominator is the log of the original base.
log(x²) = 2log(x) for all real values of x.
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Answer: False
log(x²) = 2log(x) only when x > 0. For x < 0, log(x) is undefined but log(x²) = log of a positive = defined. The rule applies only for x > 0.
Which is the inverse function of f(x) = 10ˣ?
1. f⁻¹(x) = x/10
2. f⁻¹(x) = log(x)
3. f⁻¹(x) = 10/x
4. f⁻¹(x) = xˡ⁰
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Answer: f⁻¹(x) = log(x)
f(x) = 10ˣ and log(x) = log₁₀(x) are inverses. If y = 10ˣ, then x = log(y). Swapping: f⁻¹(x) = log(x).
Condense: 3log(x) - log(y)
1. log(3x-y)
2. log(x³-y)
3. log(x³/y)
4. log(3x/y)
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Answer: log(x³/y)
3log(x) = log(x³) by the power rule. log(x³) - log(y) = log(x³/y) by the quotient rule.
The graph of y = log(x) passes through the point (1, 0).
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Answer: True
log(1) = 0 (because 10⁰ = 1), so the point (1, 0) lies on the graph of y = log(x).

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