5.4 Common and Natural
Logarithmic Functions
1. 5x=25
3. 3x=27
Do Now
Solve for x.
2. 4x=2
4. 10x=130
5.4 Common and Natural
Logarithmic Functions
1. 5x=25
x=2
3. 3x=27
x=3
Do Now
Solve for x.
2. 4x=2
x= ½
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?
Common Logarithms
• Since logs are a special kind of exponent, each
logarithmic statement can be expressed as an
exponential.
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
2. 10x = 52
log 52 = x

x = 100,000
 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
2. ex = 8
ln8 = x


x = 7.3891
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 xaxis as x-decreases
g(x) approaches the yaxis as x approaches 0
(0, 1)
(1, 0)
Reference Point
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 3
Domain: x > -4
Range: All real numbers
2. ln(2-x)-3 = ln(-(x-2))-3
Horizontal reflection across y-axis; 2 units to the
right; 3 units down
Domain: x > 2
Range: All real numbers