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MHF4U Exponential & Logarithmic Functions
Grade 12 Unit 4 — practice questions with full solutions

Log laws, converting between forms, solving exponential equations, and applications in growth, decay, pH and sound.

Free MHF4U exponential & logarithmic functions practice for Grade 12 students in Ontario. These 6 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MHF4U unit test, catching up on a lesson, or reviewing before the final exam.
MHF4U · Grade 12 Exponential and Logarithmic Functions 5 core 1 challenge
1Core
-4-3-2-112345678-4-22468xy(0, 1)(1, 0)y = 2ˣy = log₂x
Using the graph of y = 2ˣ and y = log₂x: (a) explain the relationship between them, (b) state the domain and range of each, and (c) give the asymptote of each.
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  1. (a) They are inverses of each other, so each is the reflection of the other in the line y = x.
  2. (b) For y = 2ˣ: domain is all real numbers, range is y > 0.
  3. For y = log₂x: domain and range swap, so domain is x > 0 and range is all real numbers.
  4. (c) y = 2ˣ has the horizontal asymptote y = 0; y = log₂x has the vertical asymptote x = 0.
Final answer(a) inverses, reflected in y = x   (b) 2ˣ: ℝ / y>0; log: x>0 / ℝ   (c) y = 0 and x = 0
2Core
Evaluate without a calculator: (a) log₂32   (b) log₃(1⁄27)   (c) log₅1   (d) log₂8 + log₂4
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  1. (a) Ask what power of 2 gives 32. Since 2⁵ = 32, the answer is 5.
  2. (b) 1⁄27 = 3⁻³, so log₃(3⁻³) = −3.
  3. (c) Any base to the power 0 is 1, so log of 1 is always 0.
  4. (d) Use the product law: log₂(8 × 4) = log₂32 = 5.
Final answer(a) 5   (b) −3   (c) 0   (d) 5
3Core
Write as a single logarithm:   2 log x + 3 log y − 1⁄2 log z
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  1. Use the power law to move each coefficient into the exponent: log x² + log y³ − log z1/2.
  2. Note z1/2 = √z.
  3. Use the product law on the two additions: log(x²y³).
  4. Use the quotient law for the subtraction.
Final answerlog(x²y³⁄√z)
4Core
Solve to three decimal places:   5x = 200
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  1. Take the logarithm of both sides: log(5x) = log 200.
  2. Apply the power law: x log 5 = log 200.
  3. So x = log 200 ÷ log 5.
  4. log 200 ≈ 2.30103 and log 5 ≈ 0.69897, giving x ≈ 3.29203.
Final answerx ≈ 3.292
5Core
Solve:   log₂(x + 3) + log₂(x) = 2
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  1. Combine using the product law: log₂(x(x + 3)) = 2.
  2. Convert to exponential form: x(x + 3) = 2² = 4.
  3. Expand and solve: x² + 3x − 4 = 0, which factors as (x + 4)(x − 1) = 0.
  4. So x = −4 or x = 1.
  5. Reject x = −4 because log of a negative number is undefined — always check this.
Final answerx = 1
6Challenge
An investment doubles in 9 years under continuous-style annual compounding. Find the annual growth rate as a percent, to two decimal places.
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  1. Doubling means A = 2P, so 2P = P(1 + r)⁹.
  2. Divide by P: 2 = (1 + r)⁹.
  3. Take the 9th root: 1 + r = 21/9.
  4. 21/9 ≈ 1.080060, so r ≈ 0.080060.
  5. As a percent that is about 8.01%.
Final answer≈ 8.01% per year

MHF4U Exponential & Logarithmic Functions — common questions

Short answers to the things students ask most about this unit.

What are the laws of logarithms?

The log of a product becomes a sum, the log of a quotient becomes a difference, and an exponent inside a log moves out to the front as a coefficient.

Why do you have to check solutions to logarithmic equations?

Logarithms are undefined for zero or negative arguments, so an algebraically valid answer can still be inadmissible. Substitute back and reject any that break the domain.

More MHF4U practice — grade 12 advanced functions

MHF4U Polynomial FunctionsMHF4U Polynomial Equations & InequalitiesMHF4U Rational FunctionsMHF4U Trigonometric FunctionsAll MHF4U units
← Unit 3: Rational FunctionsUnit 5: Trigonometric Functions →
Practice is step one

The classes teach the method behind every one of these.

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