Contents

Quiz 1: Powers and logs

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Quiz

This quiz tests materials in the section called “Functions, exponential and logarithm rules”. If you are struggling with the questions go back and re-read the relevant material.

Question 1: What is \(a^0\)?

  1. \(0\)

  2. \(1\)

  3. \(a\)

  4. None of the above

Question 2: What is \(3^2\)?

  1. \(6\)

  2. \(8\)

  3. \(9\)

  4. None of the above

Question 3: What is \(1^3\)?

  1. \(1\)

  2. \(3\)

  3. \(\frac{1}{3}\)

  4. None of the above

Question 4: What is \(2^{-3}\)?

  1. \(-6\)

  2. \(\frac{1}{8}\)

  3. \(-9\)

  4. None of the above

Question 5: What is \(\frac{4^3}{4^5}\)?

  1. \(4^8\)

  2. \(4^{-8}\)

  3. \(16^{-1}\)

  4. None of the above

Question 6: What is \(\left(3^{-3}\right)^3\)?

  1. \(1\)

  2. \(3^{-9}\)

  3. \(3^{-27}\)

  4. None of the above

Question 7: What is \(\frac{5^2}{3^2}\)?

  1. \(\left(\frac{5}{3}\right)^2\)

  2. \(\left(\frac{5}{3}\right)^{-2}\)

  3. \(5^{-6}\)

  4. None of the above

Question 8: What is \(4^3\)?

  1. \(12\)

  2. \(16\)

  3. \(2^6\)

  4. None of the above

Question 9: What is \(27^{-2/3}\)?

  1. \(\frac{1}{18}\)

  2. \(\frac{1}{81}\)

  3. \(\frac{1}{9}\)

  4. None of the above

Question 10: What is \(log_{10}(10^n)\)?

  1. \(10 n\)

  2. \(n\)

  3. \(10^n\)

  4. None of the above

Question 11: What is \(log_{10}\left(\frac{10^4}{10^{-3}}\right)\)?

  1. \(10^7\)

  2. \(1\)

  3. \(10\)

  4. \(7\)

Question 12: What is \(\frac{1}{2} log_{10}(16)\)?

  1. \(4\)

  2. \(8\)

  3. \(log_{10}(2)\)

  4. \(log_{10}(4)\)

Question 13: What is \(log_{10} [log_{10}(10)]\)?

  1. \(10\)

  2. \(1\)

  3. \(0\)

  4. \(-1\)

Question 14: What is \(\frac{log_{10}(1000)}{log_{10}(100)}\)?

  1. \(\frac{3}{2}\)

  2. \(1\)

  3. \(-1\)

  4. \(10\)

Question 15: Simplify the function

\[ln \left( \prod_{i=1}^n p^{x_i} (1-p)^{y_i} \right)\]

Which of these can it be written as?

  1. \(\prod_i log(p^x_i) + \prod_i log(1 - p)^y_i\)

  2. \(\sum_i x_i log(p) \times y_i log(1-p)\)

  3. \(\left(\sum_i x_i\right) log(p) + \left(\sum_i y_i \right) log(1-p)\)

  4. None of the above

Question 16: Simplify the function

\[ln \left( \prod_{i=1}^n \frac{e^{-\lambda}\lambda^{x_i}}{x_i!} \right)\]

Which of these can it be written as?

  1. \(-n \lambda + \sum_i x_i log(\lambda) - \sum_i log(x!)\)

  2. \(\prod_i \lambda + \prod_i log(\lambda^x_i) + \prod_i log\left(\frac{1}{x_i!}\right)\)

  3. \(\sum_i log(-\lambda) + \sum_i x_i log(\lambda) - \sum_i log(x!)\)

  4. None of the above

Distributions Quiz 2: Differentiation and integration