BC Calculus Assignment 35: Exponentials Worksheet A

I. Express the logarithms in terms of \( \ln{ 2} \) and \( \ln {3} \) using the rules of arithmetic for logarithms. For example, \( \ln {1.5} = \ln { \left( \frac{3}{2} \right) } = \ln {3} - \ln {2}\).
  1. \( \ln { \left( \frac{4}{9} \right) } \)
    2. Solution \( 2\ln{2} - 2\ln{3}\)
  2. \(\ln{12} \)
    3. Solution \( 2\ln{2} + \ln{3} \)
  3. \(\ln{\sqrt[3]{9}} \)
    4. Solution \( \frac{2}{3}\ln{3} \)
  4. \(\ln{(3 \sqrt{2})} \)
    5. Solution \( \ln{3} + \frac{1}{2}\ln{2} \)
II. Find \( \frac{dy}{dx}\)
  1. \( \ln \left( x^2 \right) \)
    2. Solution \( \frac{2}{x} \)
  2. \( \ln { \left( \frac{1}{x} \right) } \)
    3. Solution \( -\frac{1}{x} \)
  3. \( y = \ln {(2 - \cos{x})} \)
    4. Solution \( \frac{\sin{x}}{2 - \cos{x}} \)
  4. \( y =\ln {\left( \ln {x} \right)} \)
    5. Solution \( \frac{1}{x \ln {x}} \)
  5. \( y = x \ln{x} - x \)
    6. Solution \( \ln{x} \)
  6. \( y = \frac{x \sin{x}}{\sqrt{\sec{x}}} \)
    7. Solution \( \ln{y} = \ln{x} + \ln{ ( \sin {x} ) } - \frac{1}{2} \ln{\sec {x}}, so \frac{y'}{y} = \frac{1}{x} + \frac{\cos{x}}{\sin{x}} - \frac{1}{2} \frac{\sec {x}\tan {x} } { \sec{x}} \). \(y' = y \left( \frac{1}{x} + \cot {x} - \frac{1}{2} \tan {x} \right) \), and substitute back for \(y\).
  7. \( y = \sqrt{\frac{x}{(x + 1)(x - 3)}} \)
    8. Solution \( \ln{y} = \frac{1}{2} \left( \ln{x} - \ln{(x + 1)} - \ln{(x - 3)} \right)\), so \(\frac{y'}{y} = \frac{1}{2} \left( \frac{1}{x} - \frac{1}{x + 1} - \frac{1}{x - 3} \right) \). Then \(y' = \frac{1}{2} y \left( \frac{1}{x} - \frac{1}{x + 1} - \frac{1}{x - 3} \right) \) and substitute back for \( y\).
III. Evaluate the following integrals analytically. Support with your calculator.
  1. \( \int_{2}^{5} \frac{dx}{x} = \)
    2. Solution \( \ln{5} - \ln{2} \)
  2. \( \int_{3}^{4} \frac{dx}{2x} = \)
    3. Solution \( \frac{1}{2}(\ln{4} - \ln{3}) \)
  3. \( \int_{1}^{e} \frac{dx}{x - 4} = \)
    4. Solution \( \ln{|e - 4|} - \ln {|-3|} \)
  4. \( \int_{-1}^{0} \frac{3dx}{x + 3} = \)
    5. Solution \( 3(\ln{|3|}-\ln{|2|}) \)
  5. \( \int_{-10}^{-8} \frac{3dx}{x - 5} = \)
    6. Solution \( 3(\ln{|-13|} - \ln{|-15|}) \)