# 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)

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

### 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

### 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