Become Friends with Logarithms

Nothing like lots of practice to become proficient at this stuff! Please fill in this sheet, and practice, practice, practice!!!

After you have done each Problem Set, click on the answers button to get the correct result.

General rule: y = bx is the same as x = logby. Watch the animation below.

The three properties of logarithms are:

    1. log (xy) = log x + log y

    2. log(x/y) = log x - log y

    3. log xn = n log x

Problem Set

Set 1: Rewrite these exponentials into logarithmic form.

Example: 34 = 81 becomes log3 81 = 4

  1. 43 = 64
  2. 27 = 128
  3. 31 = 3
  4. 50 = 1
  5. 82 = 64
  6. 3-1 = 1/3
  7. 4-2 = 1/16
  8. 105 = 100,000
  9. 33 = 27
  10. e5 = 148.413

Set 2: Rewrite these logarithms into exponential form.

Example: log3 81 = 4 becomes 34 = 81
  1. log2 8 = 3
  2. log3 9 = 2
  3. log6 1/6 = -1
  4. log4 8 = 3/2
  5. log 10000 = 4
  6. ln 4 = 1.386
  7. log7 1 = 0
  8. log2 64 = 6
  9. log3 3 = 1
  10. log8 16 = 4/3

Set 3: Solve for x by rewriting into exponential format.

Example: logx 36 = 2 becomes x2 = 36, so x = 6.

  1. log2 x = 5
  2. logx 25 = 2
  3. log4 16 = x
  4. log3 x = 2
  5. logx 7 = 1
  6. log14 1 = x
  7. log8 64 = x
  8. log5 5 = x
  9. logx 2 = .5
  10. log x = 3

Set 4: Expand these logarithms:

Example: log (x3y2) = log x3 + log y2 = 3 log x + 2 log y
  1. ln (x2y)
  2. log (10xy)
  3. log (x3y/z)
  4. ln (x/y2)
  5. ln ((xy2))

Set 5: Contract these logarithms:

Example: 3 log x - 2 log y = log x3 - log y2 = log (x3/y2)

 

  1. log 4 - log x + 2 log y
  2. log x + 2 log y - log z
  3. log (x + 1) + log x
  4. ln x + 2 ln y + 3 ln z

Set 6: Using logs to solve exponential equations.

Example:

Solve for x:

    1. 7x = 17
    2. log 7x = log 17
    3. x log 7 = log 17
    4. x = (log 17)/(log 7) (using Property 3)
    5. x = 1.4559
    6. Check: 71.4559 = 17
  1. 5x = 10
  2. 4x = 18
  3. 2x = 15
  4. 3(x+1) = 15
  5. e(2+x) = 6
  6. x = log3 11 (rewrite into exponential form first!)
  7. x = log6 20
  8. x = log2 512
  9. x = ln 5
  10. 4.5x = 7.9
  11. 23e0.023x = 76
  12. 1500e1.24x = 1200

© 2004 Terry Turner, the Unknown Colleague and ASU Department of Mathematics and Statistics