1. 5.4 Common and Natural Logarithmic Functions
Do Now
Solve for x.
1. 5x= x=2
3. 3x= x=130

2. 5.4 Common and Natural Logarithmic Functions
Do Now
Solve for x.
1. 5x=25 x= x= x= ½
3. 3x=27 x= x=130 x≈2.11

3. 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?

4. Common Logarithms
Since logs are a special kind of exponent, each logarithmic statement can be expressed as an exponential.
Logarithmic
Exponential
log 29 =
= 29
log 378 =
= 378

5. Example 1: Evaluating Common Logs
Without using a calculator, find each value.
log 1000
log 1
log 10
log (-3)

6. Example 1: Solutions Without using a calculator, find each value
log 1000  10x = 1000  log 1000 = 3
log 1  10x = 1  log 1 = 0
log 10  10x = 10  log 10 = 1/2
log (-3)  10x = -3  undefined

7. 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.

8. 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 =

9. Example 2: Using Equivalent Statements
Solve each equation by using an equivalent statement.
log x = 5
10x = 52

10. Example 2: Solution
Solve each equation by using an equivalent statement.
log x = = x  x = 100,000
10x = 52 log 52 = x  x ≈

11. 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.

12. 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 =
e = 14
ln 0.2 =
e = 0.2

13. Example 3: Evaluating Natural Logs
Use a calculator to find each value
ln 1.3
ln 203
ln (-12)

14. Example 3: Solutions Use a calculator to find each value ln 1.3 .2624
ln (-12) undefined
Why is this undefined??

15. Example 4: Solving by Using and Equivalent Statement
Solve each equation by using an equivalent statement.
ln x = 2
ex = 8

16. Example 4: Solutions
Solve each equation by using an equivalent statement.
ln x = 2 e2 = x  x =
ex = 8 ln8 = x  x =

17. 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
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)

18. Example 5: Transforming Logarithmic Functions
Describe the transformation of the graph for each logarithmic function. Identify the domain and range.
3log(x+4)
ln(2-x)-3

19. Example 5: Transforming Logarithmic Functions
Describe the transformation of the graph for each logarithmic function. Identify the domain and range.
3log(x+4)
Shifted to the left 4 units; vertically stretched by 3
Domain: x > -4 Range: All real numbers
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