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Quiz: Logarithmic Functions

Test your understanding of logarithms, logarithm laws, and solving log equations with these review questions.

1. Rewrite \(\log_3 81 = 4\) in exponential form.

  1. \(4^3 = 81\)
  2. \(3^4 = 81\)
  3. \(81^3 = 4\)
  4. \(3^{81} = 4\)
Show Answer

The correct answer is B. The log-exponential conversion is: \(\log_a x = y \iff a^y = x\). Here the base is \(3\), the result (exponent) is \(4\), and the argument is \(81\). So \(3^4 = 81\). A common mistake is to swap the roles of base and argument. Remember: the base of the log becomes the base of the power, and the log's value becomes the exponent.

Concept Tested: Log-Exponential Inverse

2. What is \(\log_2 32\)?

  1. \(4\)
  2. \(5\)
  3. \(16\)
  4. \(6\)
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The correct answer is B. \(\log_2 32\) asks: "To what power must \(2\) be raised to get \(32\)?" Since \(2^5 = 32\), the answer is \(5\). Working in increments: \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\), \(2^4 = 16\), \(2^5 = 32\). Counting the powers of \(2\) up to \(32\) is a reliable way to verify.

Concept Tested: Logarithmic Function

3. Which expression equals \(\log(xy)\)?

  1. \(\log x \cdot \log y\)
  2. \((\log x)(\log y)\)
  3. \(\log x + \log y\)
  4. \(\log x - \log y\)
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The correct answer is C. The product rule of logarithms states that \(\log(mn) = \log m + \log n\) — the log of a product equals the sum of the logs. A very common mistake is to think the log of a product is the product of the logs (options A/B). Remember: logs convert multiplication into addition, which is their historic superpower for simplifying calculations.

Concept Tested: Logarithm Laws

4. Which logarithm is the natural logarithm?

  1. \(\log_{10} x\)
  2. \(\log_2 x\)
  3. \(\ln x\)
  4. \(\log x\)
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The correct answer is C. The natural logarithm uses Euler's number \(e\) as its base and is written \(\ln x\), which is shorthand for \(\log_e x\). Option A and D are both the common logarithm (base \(10\)) — in fact \(\log x\) without a subscript means \(\log_{10} x\). Option B is base \(2\), sometimes called the binary logarithm.

Concept Tested: Natural Logarithm

5. Solve \(2^x = 20\) using logarithms. Which expression gives \(x\)?

  1. \(x = \frac{\log 20}{\log 2}\)
  2. \(x = \log 20 - \log 2\)
  3. \(x = \log 20 \cdot \log 2\)
  4. \(x = \frac{20}{2}\)
Show Answer

The correct answer is A. Take the log of both sides: \(\log(2^x) = \log 20\). By the power rule, \(x \log 2 = \log 20\). Divide by \(\log 2\): \(x = \frac{\log 20}{\log 2} \approx 4.32\). This is essentially the change of base formula: \(x = \log_2 20\). Option B incorrectly applies the quotient rule; option D incorrectly treats the exponential as simple division.

Concept Tested: Solving Log Equations

6. What is the \(x\)-intercept of \(f(x) = \log_5 x\)?

  1. \((1, 0)\)
  2. \((0, 0)\)
  3. \((5, 0)\)
  4. \((0, 1)\)
Show Answer

The correct answer is A. For any base \(a\), \(\log_a 1 = 0\) because \(a^0 = 1\). So the graph of \(f(x) = \log_a x\) always crosses the \(x\)-axis at \(x = 1\), giving the point \((1, 0)\). The graph has a vertical asymptote at \(x = 0\), so there is no point on the \(y\)-axis. This is a key feature shared by all logarithmic functions.

Concept Tested: Graphing Logarithms

7. Simplify \(\log_3 54 - \log_3 6\).

  1. \(9\)
  2. \(48\)
  3. \(2\)
  4. \(3\)
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The correct answer is C. Using the quotient rule: \(\log_3 54 - \log_3 6 = \log_3 \frac{54}{6} = \log_3 9 = 2\) (since \(3^2 = 9\)). Don't subtract the arguments directly (option B gives \(48\), which is wrong). The quotient rule converts subtraction of logs into division of arguments.

Concept Tested: Logarithm Laws

8. Using change of base, express \(\log_4 50\) in terms of common logarithms.

  1. \(\log 50 \cdot \log 4\)
  2. \(\log 50 - \log 4\)
  3. \(\log 50 + \log 4\)
  4. \(\frac{\log 50}{\log 4}\)
Show Answer

The correct answer is D. The change of base rule states \(\log_a x = \frac{\log_b x}{\log_b a}\). Applying with \(a = 4\), \(x = 50\), and \(b = 10\): \(\log_4 50 = \frac{\log 50}{\log 4}\). This lets you evaluate any logarithm using a calculator's LOG button. Numerically, \(\log_4 50 \approx \frac{1.699}{0.602} \approx 2.822\).

Concept Tested: Change of Base Rule

9. A magnitude \(6\) earthquake is how many times more intense than a magnitude \(4\)?

  1. \(2\) times
  2. \(20\) times
  3. \(1000\) times
  4. \(100\) times
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The correct answer is D. The Richter scale is logarithmic: each whole-number increase represents a \(10\)-fold increase in intensity. The difference between magnitudes \(6\) and \(4\) is \(2\) units, so the intensity ratio is \(10^2 = 100\). Logarithmic scales compress huge ranges, so small differences in magnitude mean large differences in actual intensity — this is why a magnitude \(8\) quake is so catastrophic.

Concept Tested: Logarithmic Scale

10. Solve \(\log_2(x - 1) = 3\).

  1. \(x = 7\)
  2. \(x = 9\)
  3. \(x = 4\)
  4. \(x = 10\)
Show Answer

The correct answer is B. Convert to exponential form: \(\log_2(x - 1) = 3\) means \(2^3 = x - 1\), so \(8 = x - 1\) and \(x = 9\). Verify: \(\log_2(9 - 1) = \log_2 8 = 3\). ✓ Remember that log equations must be checked to ensure the argument is positive; here \(x - 1 = 8 > 0\), so the solution is valid.

Concept Tested: Solving Log Equations