MHF4U Exponential and Logarithmic Equations: Practice With Answers

To solve an exponential equation, first check whether both sides can be written with the same base. If not, logarithms may help isolate the exponent. For a logarithmic equation, state that every original logarithm argument must be positive before applying log laws or solving.

The ten original questions below practise those decisions. Unless indicated otherwise, solve over the real numbers. Logarithm bases are positive and not equal to 1; “log” without a base means base 10 in these questions.

Which rule applies?

For positive a and b, logₖ(ab) = logₖ(a) + logₖ(b), and logₖ(a/b) = logₖ(a) − logₖ(b). The power rule lets an exponent become a multiplier under its valid real-domain assumptions. There is no rule saying log(a + b) = log(a) + log(b).

An exponential statement and a logarithmic statement express the same relationship in different ways: 2³ = 8 means log₂(8) = 3. Before memorizing a rearrangement, practise explaining which quantity is the exponent.

Ten worked practice questions

1. Match the bases

Solve 2^(x + 1) = 16.

Since 16 = 2⁴, x + 1 = 4, so x = 3. Matching exponents is justified because the exponential function with base 2 is one-to-one.

2. Express both sides using a smaller common base

Solve 9^x = 27.

Write 9 = 3² and 27 = 3³. Then 3^(2x) = 3³, so 2x = 3 and x = 3/2. Matching the original numbers 9 and 27 as if their exponents had the same base would not be valid.

3. Use logarithms when there is no convenient common base

Solve 5^x = 12.

Take natural logs: x ln 5 = ln 12. Therefore x = ln 12 / ln 5 ≈ 1.54. Keep the exact quotient until the final rounding.

4. Convert a logarithmic equation

Solve log₃(x − 1) = 2.

The restriction is x > 1. Converting gives x − 1 = 3² = 9, so x = 10. It satisfies the original restriction.

5. Combine logs and reject an invalid root

Solve log₂(x) + log₂(x − 2) = 3.

Require x > 2. Combine: log₂[x(x − 2)] = 3, so x² − 2x = 8. Factoring x² − 2x − 8 gives (x − 4)(x + 2) = 0. The candidates are 4 and −2; only x = 4 satisfies x > 2.

6. Use the quotient rule

Solve log₅(x + 4) − log₅(x) = 1.

Require x > 0. Then log₅[(x + 4)/x] = 1, so (x + 4)/x = 5. Thus x + 4 = 5x and x = 1. Check: log₅5 − log₅1 = 1 − 0 = 1.

7. Substitute for a repeated exponential expression

Solve 4^x − 5(2^x) + 4 = 0.

Since 4^x = (2^x)², let u = 2^x, with u > 0. Then u² − 5u + 4 = (u − 1)(u − 4) = 0. Both positive values are allowed. From 2^x = 1 or 4, x = 0 or x = 2.

8. Use a continuous-time decay model as stated

A model gives N(t) = 800(1/2)^(t/6), where t is measured in hours. When does the model reach 100?

Divide by 800: (1/2)^(t/6) = 1/8 = (1/2)³. Therefore t/6 = 3 and t = 18 hours. This answer belongs to the supplied model, not a claim about a particular real substance.

9. Solve for a target in a growth model

A model gives P(t) = 500(1.04)^t. Find t when P(t) = 600, treating t as a real-valued time variable.

Divide by 500 and take logs: t = ln(1.2)/ln(1.04) ≈ 4.65 time units. If the question instead asked for the first whole-number period with a value of at least 600, the answer would be 5. Interpret the wording before rounding time.

10. Diagnose an invalid logarithm step

A student rewrites log(x + 3) as log(x) + log(3). Is that valid?

No. The product rule applies to log(3x), not log(x + 3). A counterexample is x = 1: log4 is not equal to log1 + log3 = log3. Checking a simple input can expose an invalid identity.

A useful error-checking routine

After solving, revisit the original equation. Check restrictions, substitute the candidate and confirm the requested rounding. For an application, state the units and whether the variable is continuous or restricted to whole periods.

If your answer differs from a worked solution, compare the exact expressions before deciding one is wrong. ln12/ln5 and log12/log5 give the same change-of-base value. Different notation can represent the same reasoning.

Get help with the reasoning step

Explore the MHF4U guide and Grade 12 math tutoring, or book a free one-hour assessment.

Curriculum reference: Ontario Grades 11–12 Mathematics. This is original practice, not an official examination.