Exercises

Precalculus for Everybody

Exponential Functions

In the exercises, from 1 to 7, find the value of the expression:

  1. \[ (81)^{1/4} \]
  2. \[ 8^{4/3} \]
  3. \[ (25)^{3/2} \]
  4. \[ (25)^{-3/2} \]
  5. \[ \left( \frac{1}{8} \right)^{-2/3} \]
  6. \[ \left( \frac{27}{16} \right)^{-1/2} \]
  7. \[ (0.01)^{-1} \]

In the exercises, from 8 to 13, simplify the expression:

  1. \[ \left( \cfrac{ \mathrm{e}^7 }{ \mathrm{e}^3 } \right)^{-1} \]
  2. \[ \cfrac{ 3^3 3^5 }{ \left( 3^4 \right)^3 } \]
  3. \[ \cfrac{ 5^{1/2} \left( 5^{1/2} \right)^5 }{5^4} \]
  4. \[ \cfrac{ 2^{-3}2^5 }{ \left( 2^4 \right)^{-3} } \]
  5. \[ \cfrac{ \left( 2^4 \right)^{1/3} }{ 16 \left( 2^{7/3} \right) } \]
  6. \[ \cfrac{ \left( 2^{1/3} 3^{2/3} \right)^3 }{ 3^{5/2} 3^{-1/2} } \]

In the exercises, from 14 to 19, solve the equation.

  1. \[ 2^{2x-1} = 8 \]
  2. \[ \left( \frac{1}{3} \right)^{x+1} = 27 \]
  3. \[ 8 \sqrt[3]{2} = 4^x \]
  4. \[ \left( 3^{2x}\, 3^2 \right)^4 = 3 \]
  5. \[ \mathrm{e}^{-6x+1} = \mathrm{e}^3 \]
  6. \[ \mathrm{e}^{x^{2}-2x} = \mathrm{e}^3 \]

In the exercises, from 20 to 28, sketch the graph of the function. Use the translation and reflection techniques in all of them, except for the exercises 25 and 27.

  1. \[ y = \mathrm{e}^{x+2} \]
  2. \[ y = -2\mathrm{e}^x +1 \]
  3. \[ y = \mathrm{e}^{-x} \]
  4. \[ y = \mathrm{e}^{-x}+2 \]
  5. \[ y = 2-\mathrm{e}^{-x} \]
  6. \[ y = 3^x \]
  7. \[ y = 3^{-x+2} \]
  8. \[ y = 4^x \]
  9. \[ y = -4^{-x-1} \]
  10. If   \(g(x)=A\mathrm{e}^{-kx}\),   \(g(0)=9\)   and   \(g(2)=5\),   find \(g(6)\).
  11. If   \(h(x)=30-P\mathrm{e}^{-kx}\),   \(h(0)=10\)   and   \(h(3)=-30\),   find \(h(12)\).
  1. \[ 3 \]
  2. \[ 16 \]
  3. \[ 125 \]
  4. \[ \frac{1}{125} \]
  5. \[ 4 \]
  6. \[ \frac{4}{3\sqrt{3}} \]
  7. \[ 100 \]
  8. \[ \mathrm{e}^{-4} = \frac{1}{ \mathrm{e}^4 } \]
  9. \[ \frac{1}{81} \]
  10. \[ \frac{1}{5} \]
  11. \[ 2^{14} \]
  12. \[ \frac{1}{32} \]
  13. \[ 2 \]
  14. \[ 2 \]
  15. \[ -4 \]
  16. \[ \frac{5}{3} \]
  17. \[ -\frac{7}{8} \]
  18. \[ -\frac{1}{3} \]
  19. \(-1\)   ó   \(3\)

  20. \(y = \mathrm{e}^{x + 2}\)

  21. \(y = -2 \mathrm{e}^x + 1\)

  22. \(y = \mathrm{e}^{-x}\)

  23. \(y = \mathrm{e}^{-x} + 2\)

  24. \(y = 2 – \mathrm{e}^{-x}\)

  1. \(y = 3^{-x + 2}\)

  1. \(y = -4^{-x – 1}\)

  2. \[ \frac{125}{81} \]
  3. \[ -1,590 \]