Exponential & Logarithmic Functions

1. Exponential & Logarithmic
Functions

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

3. • Population growth is a classic example
– Geometric growth
…
1
2 4
8 16
32 64
t = 0 1 2 3 4 5 6

4. Population Size vs. time
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6 7
Time (t)
Population
Size
(N)

5. Exponentials
f(x) = ax, a > 0
a > 1
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. -4
-2
0
2
4
6
8
-6 -4 -2 0 2 4 6 8
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 = 2.71828…, 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. 4. 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
The value of e to 15 places is
e = 2.718281828459045.
gets closer and closer to a fixed number. This
irrational number is denoted by e and is sometimes
called the Euler number.
As n gets larger and larger,
1
1
n
n
 

 
 

12. • Or  
1
0
1
lim 1 lim 1
x
x
x x
e x
x
 
 
   
 
 
 
1
1 x
x

x
2.59374246
2.70481384
2.71692393
2.71814593
2.71828047
2.71828169
e

0.1
0.01
0.001
0.0001
0.00001
0.000001
0


13. Properties of Exponents

am
 an
 am n

am
an
 amn
(am
)n
 amn

(a  b)m
 am
 bm
a
b






m

am
bm

a0
1

an

1
an

1
an
 an

a
b






n

b
a






n
a
m
n
 am
n
OR

14. x = loga b
ax = b
base

15. 15
Definition:
b
a
x
b x
a 


log
8
2
3
8
log 3
2 
 because

16. 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) = 1 2. 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)
N
a N
a

)
(
log
Rules of Logarithms
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
a
log
c
log
x
b
b

a
log
c
log
c
log
b
b
a 
log log 1
b a
a b 

19. COMMON LOGARITHMS
1. log 10 = 1
2. log 1 = 0
3. log 10x = x
4. 10log x
 x
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

20. NATURAL LOGARITHMS
1. ln e = 1
2. ln 1 = 0
3. log ex = x
4. eln x
 x
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

21. MODEL FOR EXPONENTIAL
GROWTH OR DECAY
  0
kt
A t A e

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

22. • Example: Radioactive decay
A radioactive material decays according to the law N(t)=5e-0.4t
0
1
2
3
4
5
6
0 2 4 6 8 10
Time t (months)
Number
of
grams
N
months
023
.
4
)
4
.
0
/(
6909
.
1
t
)
4
.
0
)/(
2
.
0
(
ln
t
t
4
.
0
)
2
.
0
(
ln
)
(e
ln
)
2
.
0
(
ln
e
5
/
1
e
5
1
t
4
.
0
t
4
.
0
t
4
.
0














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.

24. GRAPHS OF LOGARITHMIC FUNCTIONS

26. Derivatives
Memorize
dx
du
e
e
dx
d u
u


dx
du
a
a
a
dx
d u
u

 ln
)
(
dx
du
u
u
dx
d


1
ln
dx
du
nu
u
dx
d n
n 1


dx
du
a
u
u
dx
d
a 

ln
1
log

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

28. 28
Examples.
f (x) = ln (x 4 + 5)
f (x) = 4 ln √x
f ‘ (x) = )
5
x
(
dx
d
5
x
1 4
4


f ‘ (x) = 4
3
1
x 5
4x


2
1
x
1
4
)
x
(
'
f 
x
2

= 4 ln x 1/2
5
x
x
4
4
3


x
ln
2

1 2
1
x
2


29. 29
Example.

)
x
(
'
f
 
1
x
3
x 2
3
e
)
x
(
f 


 
3 2
x 3x 1 2
(3x 6x
f '( e )
x)
 
 
  )
1
x
3
x
(
dx
d
e 2
3
1
x
3
x 2
3





30. Differentiating Logarithmic Expressions
EXAMPLE
SOLUTION
Differentiate.
   









1
4
2
1
ln
3
2
x
x
x
x
This is the given expression.
   









1
4
2
1
ln
3
2
x
x
x
x
   
   
1
4
ln
2
1
ln
3
2



 x
x
x
x
     
1
4
ln
2
ln
1
ln
ln
3
2





 x
x
x
x
     
1
4
ln
2
ln
3
1
ln
2
ln
2
1





 x
x
x
x
Differentiate.
     










 1
4
ln
2
ln
3
1
ln
2
ln
2
1
x
x
x
x
dx
d

31. Differentiating Logarithmic Expressions
Distribute.
CONTINUED
 
   
   
 
1
4
ln
2
ln
3
1
ln
2
ln
2
1












x
dx
d
x
dx
d
x
dx
d
x
dx
d
Finish differentiating.
4
1
4
1
2
1
3
1
1
2
1
2
1








x
x
x
x
Simplify.
1
4
4
2
3
1
2
2
1






x
x
x
x

32. 32
General Derivative Rules
Power Rule General Power Rule
General Exponential Derivative Rule
1
n
n
x
n
x
dx
d 
 '
u
u
n
u
dx
d 1
n
n 

Exponential Rule
x
x
e
e
dx
d
 '
u
e
e
dx
d u
u

Log Rule
x
1
x
ln
dx
d

General Log Derivative Rule
'
u
u
1
u
ln
dx
d
