4.3 Rules of Logarithms

Definition of a Logarithmic Function

• For x > 0 and b > 0, b 1,• y = logb x is equivalent to by = x.• The function f (x) = logb x is the

logarithmic function with base b.

Location of Base and Exponent in Exponential and Logarithmic Forms

Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.

Exponent Exponent

Base Base

Text Example

Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

Solution With the fact that y = logb x means by = x,

c. log3 7 = y or y = log3 7 means 3y = 7.

a. 2 = log5 x means 52 = x.

Logarithms are exponents.Logarithms are exponents.

b. 3 = logb 64 means b3 = 64.

Logarithms are exponents.Logarithms are exponents.

Evaluatea. log2 16 b. log3 9 c. log25 5

Solution

log25 5 = 1/2 because 251/2 = 5.25 to what power is 5?c. log25 5

log3 9 = 2 because 32 = 9.3 to what power is 9?b. log3 9

log2 16 = 4 because 24 = 16.2 to what power is 16?a. log2 16

Logarithmic Expression Evaluated

Question Needed for Evaluation

Logarithmic Expression

Text Example

Basic Logarithmic Properties Involving One

• Logb b = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b).

• Logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

Inverse Properties of Logarithms

For x > 0 and b 1,

logb bx = x The logarithm with base b of b raised to a power equals that power.b logb x = x b raised to the logarithm with base b of a number equals that number.

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2 x.

4

2

8211/21/4f (x) = 2x

310-1-2x

2

4

310-1-2g(x) = log2 x

8211/21/4x

Reverse coordinates.

Text Example

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

Solution

We now plot the ordered pairs in both tables, connecting them with smooth curves. The graph of the inverse can also be drawn by reflecting the graph of f (x) = 2x over the line y = x.

-2 -1

6

2 3 4 5

5

4

3

2

-1-2

6

f (x) = 2x

f (x) = log2 x

y = x

Text Example cont.

Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx

• The x-intercept is 1. There is no y-intercept.• The y-axis is a vertical asymptote.• If b > 1, the function is increasing. If 0 < b < 1,

the function is decreasing.• The graph is smooth and continuous. It has no

sharp corners or edges.

Properties of Common Logarithms

General Properties Common Logarithms

1. logb 1 = 0 1. log 1 = 0

2. logb b = 1 2. log 10 = 1

3. logb bx = x 3. log 10x = x4. b logb x = x 4. 10 log x = x

Examples of Logarithmic Properties

log b b = 1

log b 1 = 0

log 4 4 = 1

log 8 1 = 0

3 log 3 6 = 6

log 5 5 3 = 3

2 log 2 7 = 7

Properties of Natural Logarithms

General Properties Natural Logarithms

1. logb 1 = 0 1. ln 1 = 0

2. logb b = 1 2. ln e = 1

3. logb bx = x 3. ln ex = x4. b logb x = x 4. e ln x = x

Examples of Natural Logarithmic Properties

log e e = 1

log e 1 = 0

e log e 6 = 6

log e e 3 = 3

4.3 Rules of Logarithms