This document discusses exponential and logarithmic functions with bases other than e. It defines these functions and describes their properties and applications. Examples are provided to illustrate differentiating and integrating these functions, as well as applying them to models of radioactive decay, compound interest, and exponential growth. Continuous compounding is introduced as the limiting case of compounding interest increasingly more often.

1. Logarithmic, Exponential, and
Other Transcendental Functions
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2. Bases Other Than e and
Applications
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3. 3
Objectives
 Define exponential functions that have bases other than
e.
 Differentiate and integrate exponential functions that
have bases other than e.
 Use exponential functions to model compound interest
and exponential growth.

4. 4
Bases Other than e

5. 5
Bases Other than e
The base of the natural exponential function is e. This
“natural” base can be used to assign a meaning to a
general base a.

6. 6
Bases Other than e
These functions obey the usual laws of exponents. For
instance, here are some familiar properties.
1. a0 = 1 2. axay = ax + y
3. 4. (ax)y = axy
When modeling the half-life of a radioactive sample, it is
convenient to use as the base of the exponential model.
(Half-life is the number of years required for half of the
atoms in a sample of radioactive material to decay.)

7. 7
Example 1 – Radioactive Half-Life Model
The half-life of carbon-14 is about 5715 years. A sample
contains 1 gram of carbon-14. How much will be present in
10,000 years?
Solution:
Let t = 0 represent the present time and let y represent the
amount (in grams) of carbon-14 in the sample.
Using a base of , you can model y by the equation
Notice that when t = 5715, the amount is reduced to half of
the original amount.

8. 8
Example 1 – Solution
When t = 11,430, the amount is reduced to a quarter of the
original amount, and so on.
To find the amount of carbon-14 after 10,000 years,
substitute 10,000 for t.
≈ 0.30 gram
The graph of y is shown in
Figure 5.23.
Figure 5.25
cont’d

9. 9
Bases Other than e
Logarithmic functions to bases other than e can be defined in
much the same way as exponential functions to other bases
are defined.

10. 10
Bases Other than e
Logarithmic functions to the base a have properties similar
to those of the natural logarithmic function.
1. loga 1 = 0
2. loga xy = loga x + loga y
3. loga xn = n loga x
4. loga = loga x – loga y
From the definitions of the exponential and logarithmic
functions to the base a, it follows that f(x) = ax and
g(x) = loga x are inverse functions of each other.

11. 11
Bases Other than e
The logarithmic function to the base 10 is called the
common logarithmic function. So, for common
logarithms,

12. 12
Example 2 – Bases Other Than e
Solve for x in each equation.
a. 3x = b. log2x = –4
Solution:
a. To solve this equation, you can apply the logarithmic
function to the base 3 to each side of the equation.
x = log3 3–4
x = –4

13. Example 2 – Solution cont’d
13
b. To solve this equation, you can apply the exponential
function to the base 2 to each side of the equation.
log2x = –4
x =
x =

14. 14
Differentiation and Integration

15. 15
Differentiation and Integration
To differentiate exponential and logarithmic functions to
other bases, you have three options:
(1) use the definitions of ax and loga x and differentiate
using the rules for the natural exponential and
logarithmic functions,
(2) use logarithmic differentiation, or
(3) use the following differentiation rules for bases other
than e given in the next theorem.

16. 16
Differentiation and Integration

17. 17
Example 3 – Differentiating Functions to Other Bases
Find the derivative of each function.
a. y = 2x
b. y = 23x
c. y = log10 cos x
d.

18. 18
Example 3 – Solution
a.
b.
c.
d. Before differentiating, rewrite the function using
logarithmic properties.

19. 19
Example 3 (d) – Solution (cont)
d. (cont) Next, apply Theorem 5.13 to differentiate the
function.

20. 20
Differentiation and Integration
Occasionally, an integrand involves an exponential function
to a base other than e. When this occurs, there are two
options:
(1) convert to base e using the formula ax = e(In a)x and then
integrate, or
(2) integrate directly, using the integration formula
which follows from Theorem 5.13.

21. 21
Example 4 – Integrating an Exponential Function to Another Base
Find ∫2xdx.
Solution:
∫2xdx = + C

22. 22
Differentiation and Integration

23. 23
Example 5 – Comparing Variables and Constants
a. [ee] = 0
b. [ex] = ex
c. [xe] = exe –1

24. 24
Example 5 – Comparing Variables and Constants
d.
cont’d

25. 25
Applications of Exponential
Functions

26. 26
Applications of Exponential Functions
An amount of P dollars is deposited in an account at an
annual interest rate r (in decimal form). What is the balance
in the account at the end of 1 year? The answer depends
on the number of times n the interest is compounded
according to the formula
A = P

27. 27
Applications of Exponential Functions
For instance, the result for a deposit of $1000 at 8%
interest compounded n times a year is shown in the table.

28. 28
Applications of Exponential Functions
As n increases, the balance A approaches a limit. To
develop this limit, use the following theorem.

29. 29
Applications of Exponential Functions
To test the reasonableness of
this theorem, try evaluating
for several values of x, as shown
in the table at the right.

30. 30
Applications of Exponential Functions
Given Theorem 5.15, take another look at the formula for
the balance A in an account in which the interest is
compounded n times per year. By taking the limit as n
approaches infinity, you obtain

31. 31
Applications of Exponential Functions
This limit produces the balance after 1 year of continuous
compounding. So, for a deposit of $1000 at 8% interest
compounded continuously, the balance at the end of 1 year
would be
A = 1000e0.08
≈ $1083.29.

32. 32
Applications of Exponential Functions

33. Example 6 – Comparing Continuous, Quarterly, and Monthly Compounding
33
A deposit of $2500 is made in an account that pays an
annual interest rate of 5%. Find the balance in the account
at the end of 5 years if the interest is compounded
(a) quarterly, (b) monthly, and (c) continuously.
Solution:

34. Example 6 – Solution cont’d
34