Exponential and logarithmic functions

Exponents

1. Evaluate the following quantities:
   

   (a)   \(6^{-1}\)

   (b)   \(5^{-3}\)

   (c)   \(\big(\frac{1}{3}\big)^{-2}\)

   Solutions:    (a)  \(\frac{1}{6}\);     (b)  \(\frac{1}{125}\);     (c)  \(9\)
2. Evaluate the following quantities:
   

   (a)   \(25\raise0.8em{\scriptstyle\frac{1}{2}}\)

   (b)   \(32\raise0.8em{\scriptstyle\frac{3}{5}}\)

   (c)   \(\big(\frac{1}{27}\big)\raise0.8em{\scriptstyle -\frac{4}{3}}\)

   (d)   \(\big(\frac{1}{2}\big)\raise0.8em{\scriptstyle -\frac{5}{2}}\)

   Solutions:    (a)  \(5\);     (b)  \(8\);     (c)  \(81\);     (d)  \(4\sqrt{2}\)
3. Simplify the following expressions:
   

   (a)   \(x^4\cdot(x^3)^2\)

   (b)   \({\displaystyle\frac{ab^7}{(ab^3)^2}}\)

   (c)   \(\left({\displaystyle\frac{m^8(p^3)^3}{m^2p^6}}\right)\raise1.5em{\frac{\scriptstyle 2}{\scriptstyle 3}}\)

   Solutions:    (a)  \(x^{10}\);     (b)  \(\frac{b}{a}\);     (c)  \(m^4p^2\)

Logarithms

4. Evaluate the following logarithms:
   

   (a)   \(\log_3 81\)

   (b)   \(\log_5 125\)

   (c)   \(\log_7 \frac{1}{7}\)

   (d)   \(\log\!\raise-0.8em{\frac{\scriptstyle 1}{\scriptstyle 3}} 27\)

   Solutions:    (a)  \(4\);     (b)  \(3\);     (c)  \(-1\);     (d)  \(-3\)
5. Evaluate the following quantities:
   

   (a)   \(\log_2 \sqrt{2}\)

   (b)   \(\log_8 32\)

   (c)   \(\log\!\raise-0.8em{\frac{\scriptstyle 1}{\scriptstyle 5}} \sqrt{5}\)

   Solutions:    (a)  \(\frac{1}{2}\);     (b)  \(\frac{5}{3}\);     (c)  \(-\frac{1}{2}\)
6. Use the laws of logarithms to express the following quantities as single logarithms:
   

   (a)   \(\log_2 5+\log_2 7\)

   (b)   \(\log_3 2-\log_3 5\)

   (c)   \(2\log 5-3\log 2\)

   Solutions:    (a)  \(\log_2 35\);     (b)  \(\log_3\frac{2}{5}\);     (c)  \(\log\frac{25}{8}\)
7. Write the following expressions as single logarithms:
   

   (a)   \(\log_5 x+\log_5 y - 2\log_5 z\)

   (b)   \(\log_3 a+\frac{1}{2}\log_3 b\)

   (c)   \(\log p-\log q -\frac{1}{2}\log r\)

   Solutions:    (a)  \(\log_5 \frac{xy}{z^2}\);     (b)  \(\log_3 a\sqrt{b}\);     (c)  \(\log\frac{p}{q\sqrt{r}}\)
8. Expand the following expressions (write them in terms of \(\log a,~ \log b\)):
   

   (a)   \(\log_2 a^3b^5\)

   (b)   \(\log_3\frac{\textstyle 81}{\textstyle ab^7}\)

   (c)   \({\displaystyle\log\frac{\sqrt{a}}{b}}\)

   (d)   \({\displaystyle\log\sqrt[\scriptstyle3]{\frac{a^5}{b^2}}}\)

   Solutions:    (a)  \(3\log_2 a+5\log_2 b\);     (b)  \(4-\log_3 a -7\log_3 b\);     (c)  \(\frac{1}{2}\log a-\log b\);     (d)  \(\frac{5}{3}\log a-\frac{2}{3}\log b\)
9. Write the following quantities as base-10 logarithms (use the notation \(\log_{10}a=\log a\)):
   

   (a)   \(\log_2 13\)

   (b)   \(\log_7 1000\)

   (c)   \({\displaystyle\log_5\frac{1234}{567}}\)

   Solutions:    (a)  \(\frac{\log 13}{\log 2}\);     (b)  \(\frac{3}{\log 7}\);     (c)  \(\frac{\log 1234 -\log 567}{\log 5}\)
10. Evaluate the following quantities. Write your answers to four significant digits:
    

    (a)   \(\log_{17} 6\)

    (b)   \(\log_4 2266\)

    (c)   \(\log\!\raise-0.8em{\frac{\scriptstyle 1}{\scriptstyle 3}} \sqrt{888}\)

    Solutions:    (a)  \(0,\!6324\);     (b)  \(5,\!573\);     (c)  \(-3,\!090\)

Equations

11. Solve the following equations:
    

    (a)   \(2^x=32\)

    (b)   \(5^{2x-1}=125\)

    (c)   \(7^{3x-10}=49\)

    (d)   \(10^{4x+1}=1000\)

    Solutions:    (a)  \(x=5\);     (b)  \(x=2\);     (c)  \(x=4\);     (d)  \(x=\frac{1}{2}\)
12. Solve the following equations:
    

    (a)   \(2\cdot2^x+3\cdot2^x=320\)

    (b)   \(7\cdot5^x+5^{x+1}=300\)

    (c)   \(3^x+3^{x+2}=270\)

    Solutions:    (a)  \(x=6\);     (b)  \(x=2\);     (c)  \(x=3\)
13. Solve the following equations. Round your solutions to four significant figures:
    

    (a)   \(2^x=33\)

    (b)   \(5^{x-3}=35\)

    (c)   \(3^{2x-1}=\frac{1}{2}\)

    Solutions:    (a)  \(x=5,\!044\);     (b)  \(x=5,\!209\);     (c)  \(x=0,\!1845\)
14. Solve the following equations. Round your solutions to three significant figures:
    

    (a)   \(5^x=4\cdot3^x\)

    (b)   \(3^{2x}=2\cdot7^x\)

    (c)   \(2^{3x-1}=5^{x+2}\)

    Solutions:    (a)  \(x=2,\!71\);     (b)  \(x=2,\!76\);     (c)  \(x=8,\!32\)
15. Solve the following equations:
    

    (a)   \(2^{2x}-10\cdot2^x+16=0\)

    (b)   \(3^{2x}-8\cdot3^x=9\)

    (c)   \(2\cdot25^x=9\cdot5^x+5\)

    (d)   \(4\cdot2^{2x}+3\cdot2^x=1\)

    Solutions:    (a)  \(x_1=1,~ x_2=3\);     (b)  \(x=2\);     (c)  \(x=1\);     (d)  \(x=-2\)
16. Solve the following equations:
    

    (a)   \(\log_2 x=3\)

    (b)   \(\log_5 x=4\)

    (c)   \(\log_3 (x+2) =3\)

    Solutions:    (a)  \(x=8\);     (b)  \(x=625\);     (c)  \(x=25\)
17. Solve the following equations:
    

    (a)   \(\log_x 64=3\)

    (b)   \(\log_x 9=4\)

    (c)   \(\log_x (2x+3)=2\)

    Solutions:    (a)  \(x=4\);     (b)  \(x=\sqrt{3}\);     (c)  \(x=3\)
18. Solve the following equations:
    

    (a)   \(\log_2 (3x+1)=\log_2(2x+6)\)

    (b)   \(\log_3(x-5)=\log_3(2x-7)\)

    (c)   \(\log(x^2-6)=\log(9x-20)\)

    Solutions:    (a)  \(x=5\);     (b)  no solutions, \(x\) doesn't exist;     (c)  \(x=7\)
19. Solve the following equations:
    

    (a)   \(\log_2(x+1)+\log_2(x-3)=\log_2(x^2-17)\)

    (b)   \(\log_5 (x+4)+\log_5 (x+7)=\log_5 (x+12)\)

    (c)   \(\log(x-5) + \log(2x-16)=2\log(x+4)\)

    (d)   \(2\log x=1+\log(x+20)\)

    Solutions:    (a)  \(x=7\);     (b)  \(x=-2\);     (c)  \(x=32\);     (d)  \(x=20\)

Graphs

20. Draw the following graphs:
    

    (a)   \(y=2^x\)

    (b)   \(y=3^x\)

    (c)   \(y=\left(\frac{1}{2}\right)^x\)

    (d)   \(y=e^x\)

21. Draw the following graphs:
    

    (a)   \(y=2^x+1\)

    (b)   \(y=\left(\frac{1}{3}\right)^x-2\)

    (c)   \(y=3^{x-2}\)

    (d)   \(y=4^{x+2}\)

22. Draw the following graphs:
    

    (a)   \(y=\log_2 x\)

    (b)   \(y=\log_3 x\)

    (c)   \(y=\log\!\raise-0.8em{\frac{\scriptstyle 1}{\scriptstyle 2}} x\)

    (d)   \(y=\ln x\)

23. Draw the following graphs:
    

    (a)   \(y=\log_3 x+2\)

    (b)   \(y=\log_3 (x+2)\)

    (c)   \(y=\ln (x-3)\)

24. Use GDC to draw the graph and then find all the values of \(x\) for which the given function is positive:
    

    (a)   \(y=2^x-4\)

    (b)   \(y=\ln(x+3)\)

    (c)   \(y=\log\!\raise-0.8em{\frac{\scriptstyle 1}{\scriptstyle 2}} x+2\)

    Solutions:    (a)  \(x\gt2\);     (b)  \(x\gt-2\);     (c)  \(0\lt x\lt4\)
25. Use GDC to find the points of intersection of the following two graphs (round the coordinates to two decimals):
    

    (a)   \(y=5^x,~~~ y=-x+3\)

    (b)   \(y=4^x,~~~ y=x^2-4x\)

    (c)   \(y=\log\!\raise-0.8em{\frac{\scriptstyle 1}{\scriptstyle 3}} x,~~~ y=\frac{\textstyle x-8}{\textstyle 3}\)

    Solutions:    (a)  \(P(0.56,~ 2.44)\);     (b)  \(P(-0.18,~ 0.77)\);     (c)  \(P(4.13,~ -1.29)\)

Modelling

26. A microbiologist studied a specific cell culture. He discovered that the formula   \(y=750\cdot1,\!08^x\)   models the growth of the population: \(y\) represents the number of cells at time \(x\) hours after the beginning of the experiment.
    

    (a)   Find the population at the beginning.

    (b)   Find the population after 5 hours.

    (c)   Find the population after 17 hours.

    Solutions:    (a)  750;     (b)  1102;     (c)  2775
27. A biologist studied the growth of cancer cells. In the beginning the observed culture contained 4100 cells. After 1 hour there were 10 250 cells already.
    

    (a)   Find the population after 2 hours.

    (b)   When will the population exceed 1 million cells?

    Solutions:    (a)  25 625;     (b)  after 6 hours
28. In year 1990 explorers discovered an island populated by 80 mice. When they returned to this island in 2000 the population of mice was 745.
    

    (a)   Find the population of mice in 1991 and in 1992.

    (b)   Find the growth factor.

    (c)   Find the annual growth rate in percents.

    Solutions:    (a)  100 (in 1991),   125 (in 1992);     (b)  1,25;     (c)  25%
29. The state of Rastafaria had the population of 4 371 000 in the beginning of the year 2000. The population is exponentially increasing at a rate of 1,7 %.
    

    (a)   Find the population of this state in the beginning of 2001.

    (b)   Find the population of this state in the beginning of 2025.

    Solutions:    (a)  4 445 307 (in 2001);     (b)  6 662 000 (in 2025)
30. In January 2000 the state of Sambaland had the population of 3 333 000. In January 2010 this state had the population of 4 184 000. Find the annual growth rate in this state.
    Solution:    The annual growth rate is 2,3% or 23‰.
31. In the beginning of the year 2010 the state of Corruptistan had the population of 2 377 470. Natural growth rate in this state is 13‰. Find out when will the population of this state exceed 3 millions.
    Solution:    Population will exceed 3 millions in 18 years, so it'll happen in the beginning of 2028.

 Index