Domov

Exponential and logarithmic functions

Exponents

  1. Evaluate the following quantities:

    (a)   61

    (b)   53

    (c)   (13)2

    Solutions:    (a)  16;     (b)  1125;     (c)  9
  2. Evaluate the following quantities:

    (a)   2512

    (b)   3235

    (c)   (127)43

    (d)   (12)52

    Solutions:    (a)  5;     (b)  8;     (c)  81;     (d)  42
  3. Simplify the following expressions:

    (a)   x4(x3)2

    (b)   ab7(ab3)2

    (c)   (m8(p3)3m2p6)23

    Solutions:    (a)  x10;     (b)  ba;     (c)  m4p2

Logarithms

  1. Evaluate the following logarithms:

    (a)   log381

    (b)   log5125

    (c)   log717

    (d)   log1327

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

    (a)   log22

    (b)   log832

    (c)   log155

    Solutions:    (a)  12;     (b)  53;     (c)  12
  3. Use the laws of logarithms to express the following quantities as single logarithms:

    (a)   log25+log27

    (b)   log32log35

    (c)   2log53log2

    Solutions:    (a)  log235;     (b)  log325;     (c)  log258
  4. Write the following expressions as single logarithms:

    (a)   log5x+log5y2log5z

    (b)   log3a+12log3b

    (c)   logplogq12logr

    Solutions:    (a)  log5xyz2;     (b)  log3ab;     (c)  logpqr
  5. Expand the following expressions (write them in terms of loga, logb):

    (a)   log2a3b5

    (b)   log381ab7

    (c)   logab

    (d)   loga5b23

    Solutions:    (a)  3log2a+5log2b;     (b)  4log3a7log3b;     (c)  12logalogb;     (d)  53loga23logb
  6. Write the following quantities as base-10 logarithms (use the notation log10a=loga):

    (a)   log213

    (b)   log71000

    (c)   log51234567

    Solutions:    (a)  log13log2;     (b)  3log7;     (c)  log1234log567log5
  7. Evaluate the following quantities. Write your answers to four significant digits:

    (a)   log176

    (b)   log42266

    (c)   log13888

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

Equations

  1. Solve the following equations:

    (a)   2x=32

    (b)   52x1=125

    (c)   73x10=49

    (d)   104x+1=1000

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

    (a)   22x+32x=320

    (b)   75x+5x+1=300

    (c)   3x+3x+2=270

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

    (a)   2x=33

    (b)   5x3=35

    (c)   32x1=12

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

    (a)   5x=43x

    (b)   32x=27x

    (c)   23x1=5x+2

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

    (a)   22x102x+16=0

    (b)   32x83x=9

    (c)   225x=95x+5

    (d)   422x+32x=1

    Solutions:    (a)  x1=1, x2=3;     (b)  x=2;     (c)  x=1;     (d)  x=2
  6. Solve the following equations:

    (a)   log2x=3

    (b)   log5x=4

    (c)   log3(x+2)=3

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

    (a)   logx64=3

    (b)   logx9=4

    (c)   logx(2x+3)=2

    Solutions:    (a)  x=4;     (b)  x=3;     (c)  x=3
  8. Solve the following equations:

    (a)   log2(3x+1)=log2(2x+6)

    (b)   log3(x5)=log3(2x7)

    (c)   log(x26)=log(9x20)

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

    (a)   log2(x+1)+log2(x3)=log2(x217)

    (b)   log5(x+4)+log5(x+7)=log5(x+12)

    (c)   log(x5)+log(2x16)=2log(x+4)

    (d)   2logx=1+log(x+20)

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

Graphs

  1. Draw the following graphs:

    (a)   y=2x

    (b)   y=3x

    (c)   y=(12)x

    (d)   y=ex

  2. Draw the following graphs:

    (a)   y=2x+1

    (b)   y=(13)x2

    (c)   y=3x2

    (d)   y=4x+2

  3. Draw the following graphs:

    (a)   y=log2x

    (b)   y=log3x

    (c)   y=log12x

    (d)   y=lnx

  4. Draw the following graphs:

    (a)   y=log3x+2

    (b)   y=log3(x+2)

    (c)   y=ln(x3)

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

    (a)   y=2x4

    (b)   y=ln(x+3)

    (c)   y=log12x+2

    Solutions:    (a)  x>2;     (b)  x>2;     (c)  0<x<4
  6. Use GDC to find the points of intersection of the following two graphs (round the coordinates to two decimals):

    (a)   y=5x,   y=x+3

    (b)   y=4x,   y=x24x

    (c)   y=log13x,   y=x83

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

Modelling

Use exponential growth to model the following situations.
  1. A microbiologist studied a specific cell culture. He discovered that the formula   y=7501,08x   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
  2. 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
  3. 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%
  4. 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)
  5. 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‰.
  6. 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.

Powered by MathJax
Index

 Index