BC Calculus Assignment 36: Exponentials Worksheet B



  1. Use the fact that \(y = \ln {x}\) and \(y = e^x\) are inverses of each other to simplify the following expressions:
    1. \( e^{\ln {7}} \)
    2. Solution 7
    3. \( e^{3 \ln {2 }}\)
    4. Solution 8
    5. \( e^{-2 \ln {3 }}\)
    6. Solution \(\frac{1}{9}\)
    7. \( e^{2 + \ln {3 }}\)
    8. Solution \(3e^2\)
  2. Solve the following for \(y\).
    1. \( \ln {y} = 2t + 4 \)
    2. Solution \(y = e^{2t + 4}\)
    3. \( \ln {(1 - 2y)} = t \)
    4. Solution \( y = \frac{e^t - 1}{ -2} \)
    5. \( 5 + \ln {y} = 2 ^{x^2 + 1} \)
    6. Solution \( y = e^{2^{x^2 + 1} - 5} \)
    7. \( \ln {(2^y - 1)} = x^2 - 3 \)
    8. Solution \( y = \frac{ \ln{ \left( e^{x^2 - 3} + 1 \right) }}{\ln{2}}\)
  3. Find \( \frac{dy}{dx} \).

    1. \( y = 2e^x \)
    2. Solution \( y' = 2e^x \)
    3. \( y = e^{-\frac{x}{4}} \)
    4. Solution \( y' = -\frac{1}{4} e^{-\frac{x}{4}} \)
    5. \( y = x^2e^x - xe^x \)
    6. Solution \( y' = e^x(x^2 + x - 1) \)
    7. \( y = e^{x^2} \)
    8. Solution \( y' = 2xe^{x^2} \)