Section 4.2 The Natural Exponential Function
It is known that
(
)x
1
1+
2.7182818284590452353602874713526624977572470936.
x
as x . We denote this number by e.
(
)x
1
1+
x
Value
x
1
21
2
10
1.110
2.593742460
100
1.01100
2.704813829
1000
1000
1.001
2.71692
Section 4.3 Logarithmic Functions
DEFINITION: Let a be a positive number with a = 1. The logarithmic function with base
a, denoted by loga , is dened by
loga x = y
ay = x
So, loga x is the exponent to which the base a must be raised to give x.
4
4
4
y2
y2
Section 4.6 Modeling with Exponential and Logarithmic
Functions
Many processes that occur in nature, such as population growth, radioactive decay, heat diusion, and numerous others, can be modeled using exponential functions. Logarithmic functions
are use
Section 4.5 Exponential and Logarithmic Equations
Exponential Equations
An exponential equation is one in which the variable occurs in the exponent.
EXAMPLE: Solve the equation 2x = 7.
Solution 1: We have
2x = 7
[
log2 2x = log2 7
]
[
]
x log2 2 = log2 7
Section 4.4 Laws of Logarithms
LAWS OF LOGARITHMS: If x and y are positive numbers, then
1. loga (xy ) = loga x + loga y.
()
x
2. loga
= loga x loga y.
y
3. loga (xr ) = r loga x where r is any real number.
EXAMPLES:
1. Use the laws of logarithms to evalu
Section 6.4 Inverse Trigonometric Functions and Right
Triangles
DEFINITION: The inverse sine function, denoted by sin1 x (or arcsin x), is dened to be
the inverse of the restricted sine function
sin x, x
2
2
DEFINITION: The inverse cosine function, denot
Section 4.1 Exponential Functions
DEFINITION: An exponential function is a function of the form
f (x) = ax
where a is a positive constant.
( 1 )x
x
( 10 )3
1
-3
= 103 = 1000
( 10 )2
1
-2
= 102 = 100
( 10 )1
1
-1
= 101 = 10
( 10 )0
1
0
=1
( 10 )1
1
1
= 1 =