1. Exponential & Logarithmic Functions

2. Learning Objectives
Upon completing this module, you should be able to:
 Define Logarithmic and Exponential functions .
Understand the inverse relation between Logarithmic and Exponential functions.
Know that Logarithmic and Exponential functions can be used to model a variety of real world situations.
Use the properties of logarithms and exponents for rewriting expressions and solving equations, and graphing functions
Find the derivatives of Logarithmic and Exponential functions .

3. … Population growth is a classic example Geometric growth 32 64 8 16 45 6

5. Exponentials f(x) = ax, a > 0
exponential increase
0 < a < 1
exponential decrease
As x becomes very negative, f(x) gets close to zero
As x becomes very positive, f(x) gets close to zero

6. f(x) = ax is one-to-one. For every x value there is a unique value of f(x).
This implies that f(x) = ax has an inverse.
f-1(x) = logax, logarithm base a of x.

7. f(x) = ax
f(x) = logax

8. logax is the power to which a must be raised to get x.
y = logax is equivalent to ay = x
f(f-1(x)) = alogax = x, for x > 0
f-1(f(x)) = logaax = x, for all x.
There are two common forms of the log fn.
a = 10, log10x, commonly written a simply log x
a = e = …, logex = ln x, natural log.
logax does not exist for x ≤ 0.

9. PROPERTIES OF EXPONENTIAL FUNCTIONS
Let f (x) = ax, a > 0 , a ≠ 1.
1. The domain of f (x) = ax is (–∞, ∞).
2. The range of f (x) = ax is (0, ∞); the entire graph lies above the x-axis.
3. For a > 1, Exponential Growth
(i) f is an increasing function, so the graph rises to the right.
(ii) as x → ∞, y → ∞.
(iii) as x → –∞, y → 0.

10. For 0 < a < 1, - Exponential Decay
(i) f is a decreasing function, so the graph falls to the right.
(ii) as x → – ∞, y → ∞.
(iii) as x → ∞, y → 0.
5. The graph of f (x) = ax has no x-intercepts, so it never crosses the x-axis. No value of x will cause f (x) = ax to equal 0.

11. THE VALUE OF e As n gets larger and larger,
gets closer and closer to a fixed number. This irrational number is denoted by e and is sometimes
called the Euler number.
The value of e to 15 places is
e = 11

12. Or

13. Properties of ExponentsOR

14. x = loga b
ax = b
base

15. Definition:

16. Rules of Logarithms Rules of Logarithms with Base a
If M, N, and a are positive real numbers with a ≠ 1, and x is
any real number, then
1. loga(a) = loga(1) = 0
3. loga(ax) = x 4.
5. loga(MN) = loga(M) + loga(N)
6. loga(M/N) = loga(M) – loga(N)
7. loga(Mx) = x · loga(M) 8. loga(1/N) = – loga(N)
These relationships are used to solve exponential or logarithmic equations

17. Changing the base of a logarithm
lo gac = x → c ≡ ax so
logbc ≡ logbax ≡ x·logba

18. Therefore or

19. COMMON LOGARITHMS
The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log10 x. Thus,
y = log x if and only if x = 10 y.
Applying the basic properties of logarithms
log 10 = 1
log 1 = 0
log 10x = x 19

20. NATURAL LOGARITHMS
The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. Thus,
y = ln x if and only if x = e y.
Applying the basic properties of logarithms
ln e = 1
ln 1 = 0
log ex = x 20

21. MODEL FOR EXPONENTIAL GROWTH OR DECAY
A(t) = amount at time t
A0 = A(0), the initial amount
k = relative rate of growth (k > 0) or decay (k < 0)
t = time 21

22. Example: Radioactive decay
A radioactive material decays according to the law N(t)=5e-0.4t
When does N = 1?
For what value of t does N = 1?

23. DOMAIN OF LOGARITHMIC FUNCTION
Domain of y = loga x is (0, ∞)
Range of y = loga x is (–∞, ∞)
Logarithms of 0 and negative numbers are not defined. 23

24. GRAPHS OF LOGARITHMIC FUNCTIONS24

26. Derivatives
Memorize

27. Examples. f (x) = 5 ln x. f ‘ (x) = (5)(1/x) = 5/x
f (x) = x5 ln x Note: We need the product rule.
f ‘ (x) =
(x 5 )(1/x) + (ln x)(5x 4 )
= x 4 + (ln x)(5x 4)
For you: Find dy/dx for y = x x

28. Examples. f (x) = ln (x 4 + 5) f ‘ (x) = f ‘ (x) = f (x) = 4 ln √x

29. Example.

30. Differentiating Logarithmic Expressions
EXAMPLE
Differentiate.
SOLUTION
This is the given expression.
Differentiate.

31. Differentiating Logarithmic Expressions
CONTINUED
Distribute.
Finish differentiating.
Simplify.

32. General Derivative Rules
Power Rule
General Power Rule
Exponential Rule
General Exponential Derivative Rule
Log Rule
General Log Derivative Rule