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Logarithm practice problems
A logarithm answers an exponent question. The equation logb(a) = c means bᶜ = a, with a positive base different from one and a positive argument in real-number work.
By ExamCat · Original study lesson · Published September 24, 2026
Understand the method
Translate to exponential form when a logarithm feels abstract. Product and quotient rules combine logarithms of positive quantities with the same base. They do not let you split the logarithm of a sum into a sum of logarithms.
Worked example, step by step
Solve log₂(x − 1) = 3.
- Write the domain restriction x − 1 > 0, so x > 1.
- Convert the equation: x − 1 = 2³.
- Solve x = 9.
- Check: log₂(8) = 3 and x = 9 satisfies the domain restriction.
A mistake to watch for
Logarithm equations can produce algebraic candidates outside the original domain. Check the argument of every original logarithm, especially after combining or exponentiating expressions.
Try it yourself: three questions
Write an answer before opening the explanation. For calculations, keep your units and intermediate steps; for concepts, state the reason for your choice.
1. Evaluate log₃(81).
Show answer and explanation
4, because 3⁴ = 81.
2. Evaluate log₁₀(0.01).
Show answer and explanation
−2, because 10⁻² = 0.01. A negative logarithm can have a positive argument smaller than one.
3. Is log₁₀(10 + 100) equal to log₁₀(10) + log₁₀(100)?
Show answer and explanation
No. The right side is 3, which is log₁₀(1,000), not log₁₀(110). The product rule concerns multiplication inside the logarithm.
Build your next practice session
Mark the first step where your reasoning differed from the explanation. Turn that step into a question, answer it without looking, then change one condition in the example and explain what would change. A correct guess is a reason to revisit the method.
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