Download Lecture 9 (Part 1)

Exponential and Logarithmic Functions
Exponential Functions
If a > 0 and a 6= 1, then the exponential function with base a is given by
f (x) = ax
For Example: f (x) = 2x is exponential function with base 2
g(x) = 3x is exponential function with base 3
h(x) = (1.4)x is exponential function with base
1.4
Properties of Exponents
Let a and b be positive numbers.
1. a0 = 1.
For example 20 = 1, 30 = 1 (1.5)0 = 1
2. axay = ax+y .
For Example 2x2y = 2x+y , 3x3y = 3x+y
x
3. aay = ax−y .
x
x−y
For Example 4
=
4
y
4
x
y
xy
4. (a ) = a .
For Example (5x)y = 5xy
5. (ab)x = axbx.
For Example ((2)(3))x = 2x3x
x
6. ( ab )x = abx .
x
2
2
x
For Example ( 3 ) = 3x
7. a−x = a1x .
For Example 5−x = 51x
Graphs of Exponential Functions
consider the function f (x) = 3x
x
f (x)
-4
-3
-2
-1
1
81
1
27
1
9
1
3
0
1
1
3
2
9
3 4
27 81
fHxL=3x
8
6
4
2
-2
-1
1
2
From the table and the graph of f (x) = 3x, we
can find the following:
1. The domain of f (x) is R (the set of all real
numbers)
2. The Range is (0, ∞)
3. as x approaches ∞, f (x) approaches ∞.
4. as x approaches −∞, f (x) approaches 0.
These four properties are followed by any exponential function f (x) = ax, where a > 0 ,
and a 6= 1.
Natural Exponential Function A very important exponential function is the natural exponential function defined as f (x) = ex, where e
is an irrational number, whose decimal approximation is
e ≈ 2.71828182846.
Limit Definition of e
The irrational number e is defined to be the
limit of (1 + x)1/x as x → 0. That is
lim (1 + x)1/x = e.
x→0
1 n
Or Equivalency lim (1 + ) .
n→∞
n
x
The behavior of f (x) = e is similar to f (x) =
ax, where a > 0 , and a 6= 1, see below the
graph of f (x) = ex
fHxL=ex
7
6
5
4
3
2
1
-2
-1
1
2
Example 1
A bacterial culture is growing according to the
model
1.25
y = 1+0.25e
t≥0
−0.4t ,
where y is the culture weight (in grams) and t
is the time (in hours). find the weight of the
culture after 0 hours, 1 hour, and 10 hours.
What is the limit of this model as t increases
without bound ?
solution
when t = 0
1.25
y=
= 1 grams
1+0.25e−0.4(0)
when t = 1
1.25
y=
= 1.071 grams
1+0.25e−0.4(1)
when t = 10
1.25
y=
= 1.244 grams
1+0.25e−0.4(10)
As t approaches infinity, the limit of y is
1.25
1.25
lim
=
lim
=
−0.4t
0.4t
t→∞ 1 + 0.25e
t→∞ 1 + (0.25/e
)
1.25
= 1.25 grams
1+0
Logarithmic Functions
The Logarithm Function loga x is defined as
loga x = y if and only if ay = x. a is called the
base of the logarithm. In calculus the most
useful base for logarithms is the number e
The Natural Logarithm Function
The natural logarithm function, denoted by
ln x, is defined as
ln x = y if and only if ey = x.
where ln x is loge x, that is the base is e
The definition implies that the natural logarithmic function and the natural exponential
function are inverse functions. For Example
ln 1 = 0 and
e0 = 1
ln e = 1 and
e1 = e
The graph of the function y = lnx is shown
below
y= ln
x
4
6
2
1
2
8
10
-1
-2
Inverse properties of Logarithms and Exponents
1. ln ex = x
2. eln x = x
For Example
ln e3 = 3
eln 5 = 5
Properties of Logarithms
1. ln(xy) = ln x + ln y
2. ln xy = ln x − ln y
3. ln xn = n ln x
We can use these properties to rewrite the logarithm of single quantity as a sum, difference,
or multiple of logarithms and vice versa
Example 2
Use the properties of logarithms to rewrite each
expression as a sum, difference, or multiple of
logarithms .
q
2+1
2(x+1)]
(b)
ln
x
(c)
ln[x
(a) ln 10
9
Solution
(a) ln 10
9 = ln 10 − ln 9
q
1 ln(x2 + 1)
(b) ln x2 + 1 = ln(x2 + 1)1/2 = 2
(c) ln[x2(x + 1)] = ln x2 + ln(x + 1) = 2 ln x +
ln(x + 1)
Example 3
Use the properties of logarithms to rewrite each
expression as the logarithm of single quantity
(a) ln x + 2 ln y
(b) 2 ln(x + 2) − 3 ln x
Solution
(a) ln x + 2 ln y = ln x + ln y 2 = ln(xy 2)
(b) 2 ln(x + 2) − 3 ln x = ln(x + 2)2 − ln x3 =
(x+2)2
ln x3
Solving Exponential and Logarithmic Equations
Example 4
Solve each equation
(a) 10 + e0.1x = 14
(b) 3 + 2 ln x = 7
Solution
(a) Moving 10 to the right side , we get,
e0.1x = 14 − 10 = 4
Take ln for each side, we get
ln e0.1x = ln 4
From the property ln ex = x , we get ,
0.1x = ln 4
4 = 10 ln 4
x = ln
0.1
(b) Moving 3 to the right side, we get
2 ln x = 4
ln x = 2
Exponentiate each side, we get
eln x = e2
x = e2
Trigonometric Functions