1. 7
INVERSE FUNCTIONS

2. Logarithmic Functions
INVERSE FUNCTIONS
7.3
Logarithmic Functions
In this section, we will learn about:
Logarithmic functions and natural logarithms.

3. If a > 0 and a ≠ 1, the exponential function
LOGARITHMIC FUNCTIONS
If a > 0 and a ≠ 1, the exponential function
f(x) = ax is either increasing or decreasing,
so it is one-to-one.
Thus, it has an inverse function f -1, which
is called the logarithmic function with base a
and is denoted by loga.

4. If we use the formulation of an inverse function given by (7.1.3),
LOGARITHMIC FUNCTIONS
Definition 1
If we use the formulation of an inverse
function given by (7.1.3),
then we have:

5. Thus, if x > 0, then logax is the exponent
LOGARITHMIC FUNCTIONS
Thus, if x > 0, then logax is the exponent
to which the base a must be raised
to give x.

6. LOGARITHMIC FUNCTIONS
Example 1
Evaluate:
log381
log255
log

7. log381 = 4 since 34 = 81 log255 = ½ since 251/2 = 5
LOGARITHMIC FUNCTIONS
Example 1
log381 = since 34 = 81
log255 = ½ since 251/2 = 5
log = since 10-3 = 0.001

8. LOGARITHMIC FUNCTIONS
Definition 2
The cancellation equations (Equations 4 in Section 7.1), when applied to the functions f(x) = ax and f -1(x) = logax, become:

9. The logarithmic function loga has domain and range .
LOGARITHMIC FUNCTIONS
The logarithmic function loga has domain and range .
It is continuous since it is the inverse of a continuous function, namely, the exponential function.
Its graph is the reflection of the graph of y = ax about the line y = x.

10. The figure shows the case where a > 1.
LOGARITHMIC FUNCTIONS
The figure shows the case where
a > 1.
The most important logarithmic functions have base a > 1.

11. The fact that y = ax is a very rapidly
LOGARITHMIC FUNCTIONS
The fact that y = ax is a very rapidly
increasing function for x > 0 is reflected in the
fact that y = logax is a very slowly increasing
function for x > 1.

12. The figure shows the graphs of y = logax
LOGARITHMIC FUNCTIONS
The figure shows the graphs of y = logax
with various values of the base a > 1.
Since loga1 = 0, the graphs of all logarithmic functions pass through the point (1, 0).

13. LOGARITHMIC FUNCTIONS
The following theorem summarizes the properties of logarithmic functions.

14. PROPERTIES OF LOGARITHMS
Theorem 3
If a > 1, the function f(x) = logax is a one-to-one, continuous, increasing function with domain (0, ∞) and range .
If x, y > 0 and r is any real number, then

15. PROPERTIES OF LOGARITHMS
Properties 1, 2, and 3 follow from the corresponding properties of exponential functions given in Section 7.2

16. Use the properties of logarithms in Theorem 3 to evaluate:
Example 2
Use the properties of logarithms in Theorem 3 to evaluate:
log42 + log432
(b) log280 - log25

17. Using Property 1 in Theorem 3, we have:
PROPERTIES OF LOGARITHMS
Example 2 a
Using Property 1 in Theorem 3, we have:
This is because 43 = 64.

18. Using Property 2, we have:
PROPERTIES OF LOGARITHMS
Example 2 b
Using Property 2, we have:
This is because 24 = 16.

19. limits of logarithmic functions.
LIMITS OF LOGARITHMS
The limits of exponential functions given in Section 7.2 are reflected in the following
limits of logarithmic functions.
Compare these with this earlier figure.

20. LIMITS OF LOGARITHMS
Equation 4
If a > 1, then
In particular, the y-axis is a vertical asymptote of the curve y = logax.

21. Hence, by Equation 4 with a = 10 > 1, we have:
LIMITS OF LOGARITHMS
Example 3
As x → 0, we know that t = tan2x → tan20 = 0 and the values of t are positive.
Hence, by Equation 4 with a = 10 > 1, we have:

22. Of all possible bases a for logarithms,
NATURAL LOGARITHMS
Of all possible bases a for logarithms,
we will see in Chapter 3 that the most
convenient choice of a base is the number e,
which was defined in Section 7.2.

23. The logarithm with base e is called
NATURAL LOGARITHMS
The logarithm with base e is called
the natural logarithm and has a special
notation:

24. If we put a = e and replace loge with ‘ln’
NATURAL LOGARITHMS
Definitions 5 and 6
If we put a = e and replace loge with ‘ln’
in (1) and (2), then the defining properties of
the natural logarithm function become:

25. In particular, if we set x = 1, we get:
NATURAL LOGARITHMS
In particular, if we set x = 1,
we get:

26. Find x if ln x = 5. From (5), we see that ln x = 5 means e5 = x
NATURAL LOGARITHMS
E. g. 4—Solution 1
Find x if ln x = 5.
From (5), we see that ln x = 5 means e5 = x
Therefore, x = e5.

27. If you have trouble working with the ‘ln’
NATURAL LOGARITHMS
E. g. 4—Solution 1
If you have trouble working with the ‘ln’
notation, just replace it by loge.
Then, the equation becomes loge x = 5.
So, by the definition of logarithm, e5 = x.

28. Start with the equation ln x = 5.
NATURAL LOGARITHMS
E. g. 4—Solution 2
Start with the equation ln x = 5.
Then, apply the exponential function to both
sides of the equation: eln x = e5
However, the second cancellation equation in Equation 6 states that eln x = x.
Therefore, x = e5.

29. Solve the equation e5 - 3x = 10.
NATURAL LOGARITHMS
Example 5
Solve the equation e5 - 3x = 10.
We take natural logarithms of both sides of the equation and use Definition 9:
As the natural logarithm is found on scientific calculators, we can approximate the solution— to four decimal places: x ≈

30. Express as a single logarithm. NATURAL LOGARITHMS Example 6
Using Properties 3 and 1 of logarithms, we have:

31. The following formula shows that logarithms with any base can be
NATURAL LOGARITHMS
The following formula shows that
logarithms with any base can be
expressed in terms of the natural
logarithm.

32. For any positive number a (a ≠ 1), we have:
CHANGE OF BASE FORMULA
Formula 7
For any positive number a (a ≠ 1),
we have:

33. Let y = logax. CHANGE OF BASE FORMULA Proof
Then, from (1), we have ay = x.
Taking natural logarithms of both sides of this equation, we get y ln a = ln x.
Therefore,

34. Scientific calculators have a key for natural logarithms.
So, Formula 7 enables us to use a calculator to compute a logarithm with any base—as shown in the following example.
Similarly, Formula 7 allows us to graph any logarithmic function on a graphing calculator or computer.

35. Evaluate log8 5 correct to six decimal places.
NATURAL LOGARITHMS
Example 7
Evaluate log8 5 correct to six
decimal places.
Formula 7 gives:

36. The graphs of the exponential function y = ex
NATURAL LOGARITHMS
The graphs of the exponential function y = ex
and its inverse function, the natural logarithm
function, are shown.
As the curve y = ex crosses the y-axis with a slope of 1, it follows that the reflected curve y = ln x crosses the x-axis with a slope of 1.

37. In common with all other logarithmic functions
NATURAL LOGARITHMS
In common with all other logarithmic functions
with base greater than 1, the natural
logarithm is a continuous, increasing function
defined on and the y-axis is
a vertical asymptote.

38. If we put a = e in Equation 4, then we have these limits:
NATURAL LOGARITHMS
Equation 8
If we put a = e in Equation 4, then we have these limits:

39. Sketch the graph of the function y = ln(x - 2) -1.
NATURAL LOGARITHMS
Example 8
Sketch the graph of the function
y = ln(x - 2) -1.
We start with the graph of y = ln x.

40. NATURAL LOGARITHMS
Example 8
Using the transformations of Section 1.3, we shift it 2 units to the right—to get the graph of y = ln(x - 2).

41. Notice that the line x = 2 is a vertical asymptote since:
NATURAL LOGARITHMS
Example 8
Then, we shift it 1 unit downward—to get the graph of y = ln(x - 2) -1.
Notice that the line x = 2 is a vertical asymptote since:

42. We have seen that ln x → ∞ as x → ∞.
NATURAL LOGARITHMS
We have seen that ln x → ∞ as x → ∞.
However, this happens very slowly.
In fact, ln x grows more slowly than any positive power of x.

43. To illustrate this fact, we compare
NATURAL LOGARITHMS
To illustrate this fact, we compare
approximate values of the functions
y = ln x and y = x½ = in the table.

44. We graph the functions here.
NATURAL LOGARITHMS
We graph the functions here.
Initially, the graphs grow at comparable rates.
Eventually, though, the root function far surpasses the logarithm.

45. NATURAL LOGARITHMS
In fact, we will be able to show in Section 7.8 that: for any positive power p.
So, for large x, the values of ln x are very small compared with xp.