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 functionsFor U and V real positive numbers:

Log_{b}1 = 0

Log_{b}(UV) = Log_{b}U + Log_{b}V
example: Log_{2}(4*8) = Log_{2}4 + Log_{2}8

Log_{b}(U/V) = Log_{b}U  Log_{b}V
example: Log_{2}(8/4) = Log_{2}8  Log_{2}4

Log_{b}U^{r} = rLog_{b}U
example: Log_{3}9^{2} = 2Log_{3}9

Log_{b}b^{x} = x
example: Log_{5}5^{2} = 2

b^{Logbx} = x , x > 0

Log_{b}U = Log_{b}V is equivalent to U = V
Examples with SolutionsExample 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*Log_{2}(x + b)
Solution to Example 1

Use the fact that f(0) = 0 to obtain
0 = a*Log_{2}(0 + b)

Divide both sides by a to obtain
Log_{2}b = 0

Solve for b
b = 2^{0}
= 1

Function f can be written as
a*Log_{2}(x + 1)

Use the fact that f(1) = 2 to obtain
2 = a*Log_{2}(1 + 1)

Simplify, Log_{2}2 = 1
a = 2

Function f is given by
f(x) = 2*Log_{2}(x + 1)
Check answer
f(0) = 2*Log_{2}(0 + 1)
= 2*Log_{2}(1)
=0
f(1) = 2*Log_{2}(1 + 1)
= 2*Log_{2}(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*Log_{3}(bx)
Example 2
Find the domain of function f where f is a logarithmic function given by
f(x) = 2 Log_{2}(2 x  4)
Solution to Example 2

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

Solve the above inequality to obtain
x > 2

Use the applet here to check your answer.
Example 3
Find all values of x for which f(x) given by
f(x) = 2 Log_{2}(2 x  4)
is positive.
Solution to Example 3

f(x) is positive if the argument of the Log_{2}(2 x  4) is greater than 1
2x  4 > 1

Solve the above inequality to obtain
x > 2.5

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) = Log_{2}(2 x + 6)
Solution to Example 4

The x intercept is found in solving Log_{2}(2 x + 6) = 0
2x + 6 = 1

Solve the above equation to obtain
x =  2.5

Use the applet here to check your answer.
More References and Links to logarithmic FunctionsLogarithmic 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). 