5.4 Common and Natural Logarithmic Functions

Do NowSolve for x.

1. 5x=25 2. 4x=2

3. 3x=27 4. 10x=130

5.4 Common and Natural Logarithmic Functions

Do NowSolve for x.

1. 5x=25 x=2 2. 4x=2 x= ½

3. 3x=27 x=3 4. 10x=130 x≈2.11

Common Logarithms• The inverse function of the exponential function f(x)=10x is

called the common logarithmic function.– Notice that the base is 10 – this is specific to the “common”

log• The value of the logarithmic function at the number x is

denoted as f(x)=log x.• The functions f(x)=10x and g(x)=log x are inverse functions.• log v = u if and only if 10u = v– Notice that the base is “understood “to be 10.

• Because exponentials and logarithms are inverses of one another, what do we know about their graphs?

• Since logs are a special kind of exponent, each logarithmic statement can be expressed as an exponential.

Common Logarithms

Logarithmic Exponential

log 29 = 1.4624 101.4624 = 29

log 378 = 2.5775 102.5775 = 378

Example 1: Evaluating Common Logs

• Without using a calculator, find each value.

1. log 1000

2. log 1

3. log 10

4. log (-3)

Example 1: Solutions

• Without using a calculator, find each value

1. log 1000 10x = 1000 log 1000 = 3

2. log 1 10x = 1 log 1 = 0

3. log 10 10x = 10 log 10 = 1/2

4. log (-3) 10x = -3 undefined

Evaluating Logarithms

• A calculator is necessary to evaluate most logs, but you can get a rough estimate mentally.

• For example, because log 795 is greater than log 100 = 2 and less than log 1000 = 3, you can estimate that log 795 is between 2 and 3, and closer to 3.

Using Equivalent Statements• A method for solving logarithmic or

exponential equations is to use equivalent exponential or logarithmic statements.

• For example:– To solve for x in log x = 2, we can use 102 = x and

see that x = 100– To solve for x in 10x = 29, we can use log 29 = x,

and using a calculator to evaluate shows that x = 1.4624

Example 2: Using Equivalent Statements

• Solve each equation by using an equivalent statement.

1. log x = 5

2. 10x = 52

Example 2: Solution

• Solve each equation by using an equivalent statement.

1. log x = 5 105 = x x = 100,000

2. 10x = 52 log 52 = x x ≈ 1.7160

Natural Logarithms• The exponential function f(x)=ex is useful in

science and engineering. Consequently, another type of logarithm exists, where the base is e instead of 10.

• The inverse function of the exponential function f(x)=ex is called the natural logarithmic function.

• The value of this function at the number x is denoted as f(x)=ln x and is called the natural logarithm.

Natural Logarithms• The functions f(x)=ex and g(x)=ln x are inverse functions.• ln v = u if and only if eu = v• Notice that the base is “understood” to be e.

• Again, as with common logs, every natural logarithmic statement is equivalent to an exponential statement.

Logarithmic Exponential

ln 14 = 2.6391 e2.6391 = 14

ln 0.2 = -1.6094 e-1.6094 = 0.2

Example 3: Evaluating Natural Logs

• Use a calculator to find each value

1. ln 1.3

2. ln 203

3. ln (-12)

Example 3: Solutions

• Use a calculator to find each value

1. ln 1.3 .2624

2. ln 203 5.3132

3. ln (-12) undefined

Why is this undefined??

Example 4: Solving by Using and Equivalent Statement

• Solve each equation by using an equivalent statement.

1. ln x = 2

2. ex = 8

Example 4: Solutions

• Solve each equation by using an equivalent statement.

1. ln x = 2 e2 = x x = 7.3891

2. ex = 8 ln8 = x x = 2.0794

Graphs of Logarithmic Functions• The following table compares graphs of exponential

and logarithmic functions (page 359 in your text):

Exponential Functions Logarithmic Functions

Examples f(x) = 10x; f(x) = ex g(x) = log x; g(x) = ln x

Domain All real numbers All positive real numbers

Range All positive real numbers All real numbers

f(x) increases as x increases

g(x) increases as x increases

f(x) approaches the x-axis as x-decreases

g(x) approaches the y-axis as x approaches 0

Reference Point (0, 1) (1, 0)

Example 5: Transforming Logarithmic Functions

• Describe the transformation of the graph for each logarithmic function. Identify the domain and range.

1. 3log(x+4)

2. ln(2-x)-3

Example 5: Transforming Logarithmic Functions

• Describe the transformation of the graph for each logarithmic function. Identify the domain and range.

1. 3log(x+4)Shifted to the left 4 units; vertically stretched by 3Domain: x > -4 Range: All real numbers

2. ln(2-x)-3 = ln(-(x-2))-3 Horizontal reflection across y-axis; 2 units to theright; 3 units downDomain: x > 2 Range: All real numbers