Tutorial on Logarithmic Functions (1)

This is a tutorial on logarithmic functions to further understand the properties of the these functions. Detailed solutions of examples and explanations are included.

Properties of the Logarithmic functions

For U and V real positive numbers:

  1. Logb1 = 0
  2. Logb(UV) = LogbU + LogbV
      example: Log2(4*8) = Log24 + Log28

  3. Logb(U/V) = LogbU - LogbV
      example: Log2(8/4) = Log28 - Log24

  4. LogbUr = rLogbU
      example: Log392 = 2Log39

  5. Logbbx = x
      example: Log552 = 2

  6. bLogbx = x , x > 0
      example: 2Log25 = 5

  7. LogbU = LogbV    is equivalent to    U = V



Examples with Solutions

Example 1

Find parameters a and b so that f(0) = 0 and f(1) = 2, where f is a logarithmic function given by
f(x) = a*Log2(x + b)

Solution to Example 1

  1. Use the fact that f(0) = 0 to obtain
      0 = a*Log2(0 + b)

  2. Divide both sides by a to obtain
      Log2b = 0

  3. Solve for b
      b = 20
      = 1


  4. Function f can be written as
      a*Log2(x + 1)


  5. Use the fact that f(1) = 2 to obtain
      2 = a*Log2(1 + 1)

  6. Simplify, Log22 = 1
      a = 2

  7. Function f is given by
      f(x) = 2*Log2(x + 1)

Check answer
f(0) = 2*Log2(0 + 1)
= 2*Log2(1)
=0
f(1) = 2*Log2(1 + 1)
= 2*Log2(2)
= 2

Matched Exercise 1
Find parameters a and b so that f(1/3) = 0 and f(1) = -2, where f is an logarithmic function given by
f(x) = a*Log3(bx)

Example 2

Find the domain of function f where f is a logarithmic function given by
f(x) = 2 Log2(2 x - 4)

Solution to Example 2

  1. To obtain the domain of f, we need to have the argument 2x - 4 positive
      2x - 4 > 0

  2. Solve the above inequality to obtain
      x > 2

  3. Use the applet here to check your answer.

Example 3

Find all values of x for which f(x) given by
f(x) = 2 Log2(2 x - 4)

is positive.

Solution to Example 3

  1. f(x) is positive if the argument of the Log2(2 x - 4) is greater than 1
      2x - 4 > 1

  2. Solve the above inequality to obtain
      x > 2.5

  3. Use the applet here to check your answer.

Example 4

Find the x intercept of the graph of f where f is a function given by
f(x) = Log2(2 x + 6)

Solution to Example 4

  1. The x intercept is found in solving Log2(2 x + 6) = 0
      2x + 6 = 1

  2. Solve the above equation to obtain
      x = - 2.5

  3. Use the applet here to check your answer.


More References and Links to logarithmic Functions

Logarithmic Functions (interactive tutorial).
Graph Logarithmic Functions.
Tutorial on Exponential and Logarithmic Equations.
Graph of Logarithmic and Exponential Functions - Self Test.
Solve Exponential and Logarithmic Equations (self test).