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Grade 12 · MHF4U · Advanced Functions

MHF4U Exponential and Logarithmic Functions Practice Test (with Solutions)

Exponential and Logarithmic Functions is the unit where logs show up as the inverse of exponentials. Expect to simplify with the log laws, find the inverse of an exponential, read the domain and asymptote of a transformed graph and solve exponential equations exactly. Work each question below on paper first, then open the answer.

Free Exponential & Logarithmic Functions Practice Test

8 questions, exactly as a free account gets them in the first Exponential & Logarithmic Functions test on Martian Lab. Try each one on paper first, then open the answer.

  1. Simplify the logarithm log⁡aan\log_a a^n
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    nn
  2. Simplify the logarithm 2log⁡2(3x+4)2^{\log_2(3x+4)}
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    3x+43x+4
  3. The restriction on log⁡2(x+2)+log⁡2(x−2)\log_2(x+2) + \log_2(x-2) is
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    x>2x > 2, x∈Rx \in \mathbb{R}
  4. The equation of the inverse of y=4xy = 4^x can be written as
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    x=4yx = 4^y or y=log⁡4xy = \log_4 x
  5. Simplify log⁡a27+log⁡a4−log⁡a12\log_a 27 + \log_a 4 - \log_a 12 (exact value, no decimals)
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    log⁡a9\log_a 9
  6. Given 43x−2=8x+14^{3x-2} = 8^{x+1}, solve for xx. Express your answer as an exact value.
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    x=73x = \dfrac{7}{3}
  7. The domain for y=−3log⁡2(x−3)+4y = -3\log_2(x-3) + 4 is
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    x>3x > 3, x∈Rx \in \mathbb{R}
  8. The equation of the asymptote for the *inverse* of y=2x+3−4y = 2^{x+3} - 4 is
    Show answer
    x=−4x = -4

This Unit on Martian Lab

The lessons start with what a logarithm is, move through graphing exponential and log functions and solving their equations, and finish with applications.

  1. Logarithmic & Exponential Functions
  2. Graphing Exponential & Logarithmic Functions
  3. Logarithmic & Exponential Equations
  4. Applications of Logarithmic & Exponential Functions

Then 4 full Exponential & Logarithmic Functions practice tests to check yourself.

Start Exponential & Logarithmic Functions free

Exponential & Logarithmic Functions FAQ

What is on an MHF4U exponential and logarithmic functions test?+

Switching between exponential and log form, the log laws, inverses of exponential functions, the domain, range and asymptotes of transformed graphs, solving exponential and log equations, and word problems. The free questions on this page cover the log laws, inverses, domains, asymptotes and exponential equations.

What are the log laws?+

Product: log_a(xy) = log_a x + log_a y. Quotient: log_a(x/y) = log_a x - log_a y. Power: log_a(x^n) = n log_a x. Together they let you collapse an expression like log_a 27 + log_a 4 - log_a 12 into log_a 9.

How do you solve an exponential equation with different bases?+

If both bases are powers of the same number, rewrite them and set the exponents equal. For 4^(3x - 2) = 8^(x + 1), both are powers of 2, so 6x - 4 = 3x + 3 and x = 7/3. If there is no common base, take the log of both sides.

How do you find the domain of a log function?+

The argument of every log has to be positive. For y = -3 log_2(x - 3) + 4 that means x > 3. When two logs are added, both arguments must be positive, so log_2(x + 2) + log_2(x - 2) needs x > 2.

How are exponential and log graphs related?+

They are inverses, so each is the other reflected in y = x. A horizontal asymptote on the exponential becomes a vertical asymptote on the log: y = 2^(x + 3) - 4 has y = -4, so its inverse has x = -4.

Exponential & Logarithmic Functions Topics

One idea at a time: a short explanation, then five worked examples.

More MHF4U Units

Previous unit: Trigonometric Functions

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