1. 7
INVERSE FUNCTIONS

2. General Logarithmic and Exponential Functions
INVERSE FUNCTIONS
7.4*
General Logarithmic and Exponential Functions
In this section, we:
Use the natural exponential and logarithmic functions
to study exponential and logarithmic functions
with base a > 0.

3. GENERAL EXPONENTIAL FUNCTIONS
If a > 0 and r is any rational number, then, by Equations 4 and 7 in Section 7.3*, ar = (eln a)r = er ln a

4. Thus, even for irrational numbers x, we define: ax = ex ln a
GENERAL EXP. FUNCTIONS
Definition 1
Thus, even for irrational numbers x, we define: ax = ex ln a
Thus, for instance,

5. The function f(x) = ax is called the exponential function with base a.
GENERAL EXP. FUNCTIONS
The function f(x) = ax is called the exponential function with base a.
Notice that ax is positive for all x because ex is positive for all x.

6. Definition 1 allows us to extend one of the laws of logarithms.
GENERAL EXP. FUNCTIONS
Definition 1 allows us to extend one of the laws of logarithms.
We know that ln(ar) = r ln a when r is rational.
However, if we now let r be any real number we have, from Definition 1, ln ar = ln(er ln a) = r ln a

7. Hence, we have: ln ar = r ln a for any real number r.
GENERAL EXP. FUNCTIONS
Equation 2
Hence, we have: ln ar = r ln a for any real number r.

8. GENERAL EXP. FUNCTIONS
The general laws of exponents follow from Definition 1 together with the laws of exponents for ex.

9. LAWS OF EXPONENTS
Laws 3
If x and y are real numbers and a, b > 0, then 1. ax+y = axay 2. ax-y = ax/ay 3. (ax)y = axy 4. (ab)x = axbx

10. Using Definition 1 and the laws of exponents for ex, we have:
LAW 1 OF EXPONENTS
Proof
Using Definition 1 and the laws of exponents for ex, we have:

11. Using Equation 2, we obtain:
LAW 3 OF EXPONENTS
Proof
Using Equation 2, we obtain:
The remaining proofs are left as exercises.

12. GENERAL EXP. FUNCTIONS
Formula 4
The differentiation formula for exponential functions is also a consequence of Definition 1:

13. GENERAL EXP. FUNCTIONS
Formula 4—Proof

14. GENERAL EXP. FUNCTIONS
Notice that, if a = e, then ln e = 1 and Formula 4 simplifies to a formula we already know: (d/dx) ex = ex
In fact, the reason the natural exponential function is used more often than other exponential functions is that its differentiation formula is simpler.

15. GENERAL EXP. FUNCTIONS
Example 1
In Example 6 in Section 3.7, we considered a population of bacteria cells in a homogeneous nutrient medium.
We showed that, if the population doubles every hour, then the population after t hours is: n = n02t where n0 is the initial population.

16. Now, we can use Formula 4 to compute the growth rate:
GENERAL EXP. FUNCTIONS
Example 1
Now, we can use Formula 4 to compute the growth rate:
For instance, if the initial population is n0 = 1000 cells, then the growth rate after two hours is:

17. Combining Formula 4 with the Chain Rule, we have:
GENERAL EXP. FUNCTIONS
Example 2
Combining Formula 4 with the Chain Rule, we have:

18. So, (d/dx) ax = ax ln a > 0.
EXPONENTIAL GRAPHS
If a > 1, then ln a > 0.
So, (d/dx) ax = ax ln a > 0.
This shows that y = ax is increasing.
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19. If 0 < a < 1, then ln a < 0.
EXPONENTIAL GRAPHS
If 0 < a < 1, then ln a < 0.
So, y = ax is decreasing.
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20. EXPONENTIAL GRAPHS
Notice from this figure that, as the base a gets larger, the exponential function grows more rapidly (for x > 0).
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21. EXPONENTIAL GRAPHS
Here, we see how the exponential function y = 2x compares with the power function y = x2.
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22. The graphs intersect three times.
EXPONENTIAL GRAPHS
The graphs intersect three times.
Ultimately, though, the exponential curve y = 2x grows far more rapidly than the parabola y = x2.
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© Thomson Higher Education

23. EXPONENTIAL GRAPHS
In Section 7.5, we will show how exponential functions occur in the description of population growth and radioactive decay.
Let’s look at human population growth.

24. The figure shows the related scatter plot.
EXPONENTIAL GRAPHS
The table shows data for the population of the world in the 20th century.
The figure shows the related scatter plot.
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25. EXPONENTIAL GRAPHS
The pattern of the data points in the figure suggests exponential growth.
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26. EXPONENTIAL GRAPHS
Hence, we use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model P = ( ) . ( )t

27. EXPONENTIAL GRAPHS
The figure shows the graph of the exponential function together with the original data points.
We see the curve fits the data reasonably well.
The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930s.
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28. The integration formula that follows from Formula 4 is:
EXPONENTIAL INTEGRALS
The integration formula that follows from Formula 4 is:

29. EXPONENTIAL INTEGRALS
Example 3

30. THE POWER RULE VS. THE EXPONENTIAL RULE
Now that we have defined arbitrary powers of numbers, we are in a position to prove the general version of the Power Rule—as promised in Section 3.3

31. If n is any real number and f(x) = xn, then f ’(x) = nxn-1
THE POWER RULE
If n is any real number and f(x) = xn, then f ’(x) = nxn-1

32. THE POWER RULE
Proof
Let y = xn and use logarithmic differentiation: ln | y | = ln | x |n = n ln | x | x ≠ 0
Therefore,
Hence,

33. You should distinguish carefully between:
POWER RULE VS. EXP. RULE
Note
You should distinguish carefully between:
The Power Rule [(d/dx) xn = nxn-1], where the base is variable and the exponent is constant.
The rule for differentiating exponential functions [(d/dx) ax = ax ln a], where the base is constant and the exponent is variable.

34. In general, there are four cases for exponents and bases:
THE POWER RULE—CASES
In general, there are four cases for exponents and bases:

35. Differentiate: Using logarithmic differentiation, we have:
THE POWER RULE
E. g. 4—Solution 1
Differentiate:
Using logarithmic differentiation, we have:

36. Another method is to write
THE POWER RULE
E. g. 4—Solution 2
Another method is to write

37. THE POWER RULE
The figure illustrates Example 4 by showing the graphs of and its derivative.
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38. If a > 0 and a ≠ 1, then f(x) = ax is a one-to-one function.
GENERAL LOG. FUNCTIONS
If a > 0 and a ≠ 1, then f(x) = ax is a one-to-one function.
Its inverse function is called the logarithmic function with base a.
It is denoted by loga.

39. Thus, loga x = y  ay = x In particular, we see that: logex = ln x
GENERAL LOG. FUNCTIONS
Formula 5
Thus, loga x = y  ay = x
In particular, we see that: logex = ln x

40. The cancellation equations for the inverse functions logax and ax are:
GENERAL LOG. FUNCTIONS
The cancellation equations for the inverse functions logax and ax are:

41. The figure shows the case where a > 1.
GENERAL LOG. FUNCTIONS
The figure shows the case where a > 1.
The most important logarithmic functions have base a > 1.
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.
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42. GENERAL LOG. FUNCTIONS
The figure shows the graphs of y = logax with various values of the base a.
Since loga1 = 0, the graphs of all logarithmic functions pass through the point (1, 0).
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43. The laws of logarithms are similar to those for the natural logarithm.
GENERAL LOG. FUNCTIONS
The laws of logarithms are similar to those for the natural logarithm.
They can be deduced from the laws of exponents.
See Exercise 65.

44. CHANGE OF BASE FORMULA
Formula 6
This formula shows that logarithms with any base can be expressed in terms of the natural logarithm.
For any positive number a (a ≠ 1), we have:

45. Then, from Formula 5, we have: ay = x
CHANGE OF BASE FORMULA
Formula 6—Proof
Let y = logax.
Then, from Formula 5, we have: ay = x
Taking natural logarithms of both sides of this equation, we get: y ln a = ln x
Therefore,

46. Scientific calculators have a key for natural logarithms.
CHANGE OF BASE FORMULA
Scientific calculators have a key for natural logarithms.
So, Formula 6 enables us to:
Use a calculator to compute a logarithm with any base (as shown in the following example).
Graph any logarithmic function on a graphing calculator or computer (as in Exercises 14-16).

47. Evaluate log85 correct to six decimal places.
CHANGE OF BASE FORMULA
Example 5
Evaluate log85 correct to six decimal places.
Formula 6 gives:

48. Formula 6 enables us to differentiate any logarithmic function.
CHANGE OF BASE FORMULA
Formula 7
Formula 6 enables us to differentiate any logarithmic function.
Since ln a is a constant, we can differentiate as follows:

49. Using Formula 7 and the Chain Rule, we get:
CHANGE OF BASE FORMULA
Example 6
Using Formula 7 and the Chain Rule, we get:

50. CHANGE OF BASE FORMULA
From Formula 7, we see one of the main reasons that natural logarithms (logarithms with base e) are used in calculus:
The differentiation formula is simplest when a = e because ln e = 1.

51. We have shown that, if f(x) = ln x, then f’(x) = 1/x.
THE NUMBER e AS A LIMIT
We have shown that, if f(x) = ln x, then f’(x) = 1/x.
Thus, f’(1) = 1.
We now use this fact to express the number e as a limit.

52. From the definition of a derivative as a limit, we have:
THE NUMBER e AS A LIMIT
From the definition of a derivative as a limit, we have:

53. THE NUMBER e AS A LIMIT
Since f’(1) = 1, we have:
Then, by Theorem 8 in Section 2.5 and the continuity of the exponential function, we have:

54. THE NUMBER e AS A LIMIT
Formula 8
The number e as a limit:

55. Formula 8 is illustrated by:
THE NUMBER e AS A LIMIT
Formula 8 is illustrated by:
The graph of the function y = (1 + x)1/x.
A table of values for small values of x.
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© Thomson Higher Education/

56. If we put n = 1/x in Formula 8, then n → ∞ as x → 0+.
THE NUMBER e AS A LIMIT
Formula 9
If we put n = 1/x in Formula 8, then n → ∞ as x → 0+.
So, an alternative expression for e is: