1. 5.5 Base Other Than e and Application

2. After this lesson, you should be able to:
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.

3. Definition of Exponential Function to Base a
Exponential Functions
Definition of Exponential Function to Base a
The exponential function obey the usual laws of exponents.

4. Definition of Logarithmic Function to Base a
Properties of Inverse Functions

5. Differentiation and Integration
After we developed the derivatives for Natural Exponential Functions, it is quite easy to derived exponential functions with bases other than e.
Notice that when a ≠1 and a > 0
and
Taking the derivatives for both functions, we have
When f(x) = x, we also can conclude

6. Theorem 5.13 Derivatives for Bases Other Than e

7. Example 1 Find the derivative of f (x) = 10x f (x) = 53x
f (x) = log2cos x
Solution

8. Example 2 Find the derivative of
Solution Use the product rule

9. When an integrand involves an exponential function to a base
other than e, there are two options:
Convert to base e using the formula
integrate directly by using the integration formula
Example 3 Find
Solution

10. Theorem 5.14 The Power Rule for Real Exponents

11. Example 4 Find the derivative of
Solution Note that we can not simply write the derivative as
Indeed, there is no simple differentiation rule to find its derivative. We have to take the logarithm first.

12. If the interest rate is compounded annually, at the end of 3rd
Applications of Exponential Functions
Question You deposit $1500 in an account that pays 7.5% interest compounded annually. What is the balance after 3 year? How does this compare to the same investment compounded monthly and daily?
If the interest rate is compounded annually, at the end of 3rd
year, the asset value is:
A3= P(1 + r )3 = 1500 ( )3 = $
If the interest rate is compounded monthly, at the end of 3rd
year, the asset value is:
A3= P (1 + r / 12)12·3 = 1500 ( / 12)36
= $

13. If the interest rate is compounded daily, at the end of 3rd year,
the asset value is:
A3= P (1 + r / 365)365·3 = 1500 ( / 365)1095
= $
From the above question, we see that the total asset is affected by the
factor
(1 + r / 12)12·3 = [(1 + r / 12)12]3
(1 + r / 365)365·3 = [(1 + r / 365)365]3
Or in general, the factor of
(1 + r / n) n
or simply,
(1 + 1 / n) n
Mathematician Euler found that the (1 + 1 / n) n is approaching a
fixed decimal number, …, as n  

14. The number
was named after Euler as Euler number or natural base.
The numbers e and possesses some
properties:
The number en increases when the n increases with an upper bound less than 3.
2) The number en is approaching to … or approximate to as n increases.
The number e is an irrational number (non-terminating and non-repeating), like . 4)

15. Theorem 5.15 A Limit Involving e

16. You deposit P dollars in an bank account that pays interest
Applications of Exponential Functions
You deposit P dollars in an bank account that pays interest
rate of r per year. If the rate is compounded n times every
year, then the asset value at the end of t years is
If the interest rate is compounded continuously, then at the end
of t years, the asset value is:

17. Example 5 You deposit $1500 dollars in an bank account that pays 7
Example 5 You deposit $1500 dollars in an bank account that pays 7.5% per year interest compounded
a) semi-annually
b) monthly
c) continously
for 3 years. Find the balance. a) b) c)

18. Homework
5.5 P. 366 Q xxxxxxxxxxxx