Chapter 5 Exponential, Logarithmic, and Inverse Trigonometric Functions
Chapter 5 Exponential, Logarithmic, and Inverse Trigonometric Functions
Exponential Functions
If f (x) = x2, then the base x varies and the exponent 2 is a constant. The function f (x) = 2x has constant base 2 and variable exponent x. This is called an exponential
function.
x f (x) = 2x
-3 1/8
-2 1/4
-1 1/2
0
1
1
2
2
4
3
8
Now lim 2x = and lim 2x = 0
x
x-
To see that f (x) = 2x gets large quickly consider the following:
Suppose you have one penny and for the entire month of February your money doubles
each day. If I offer you $100,000 or the amount you would have after February 28, which would you take?
Day
Amount
Dollars
1
2
$0.02
7
27 = 1,024
$10.24
14
214 = 16,384
$163.84
21 221 = 2,097,152 $20,971.52
28 228 = 2.6844 ? 108 $2,684,400
Hence you can see how fast the exponential function rises.
127
Now let f (x) =
1 2
x = 2-x
x 2-x -2 4 -1 2 01 1 1/2 2 1/4
The most important exponential function is f (x) = ex. We base our development on the following definition:
lim(1 + x)1/x = e
x0
x 1
10 1
100 1
1,000 1
10,000 1
100,000
(1 + x)1/x (1.1)10 = (1.01)100 = (1.001)1,000 = (1.0001)10,000 = (1.00001)100,000 =
Do the above calculations on your calculator and find that e 2.7182818.
128
Now let f (x) = ex and complete the table of values on your calculator.
x ex 0 1 10 100 1000 -1 -10 -100
Thus the graph of f (x) = ex is
lim ex = and lim ex = 0
x
x-
If 0 < a, a = 1, then f (x) = ax is the general exponential function. All of the familiar laws of exponents hold. If 1 < a, then y = ax is increasing and if 0 < a < 1, then y = ax is decreasing. Thus, if ax1 = ax2, then x1 = x2. Also note that ax > 0.
129
Exercises on Exponential Functions
1. Graph
a. f (x) = 3x
b. f (x) =
1x 3
c. f (x) = e-x
d. f (x) = -(2-x)
2. Find the limits
a. lim (1.001)x
x
b. lim (.999)x
x
c. lim e-x
x
d. lim e-x
x-
e.
lim
x
e2x e2x
- +
e-2x e-2x
f. lim e1/(1-x)
x1+
g. lim e1/(1-x)
x1-
3. Suppose the graphs of f (x) = x2 and g(x) = 2x are drawn where the unit of
measurement is 1 in. If x = 24 in, then find f (x) in feet and g(x) in miles.
130
Logarithmic Functions
Definition: loga x = y means ay = x. loga x is read logarithm of x to the base a.
Example 1
Let y = log2 x
x log2 x 10 21 42 83 1/2 -1 1/4 -2
Remarks: log2(0) = y means 2y = 0 which is impossible. Also log2(-1) = y means 2y = -1 which is impossible. Thus the domain of y = log2 x is (0, ).
Common Logarithm
The common logarithm has base a = 10. We write y = log10 x = log x which means
10y
=
x.
Thus
log 1
=
0,
log 10
=
1,
log 100
=
2,
log
1 10
=
-1,
etc.
Natural Logarithm
The natural logarithm has base a = e. We write y = loge x = ln x which means ey = x. Complete the following table on your calculator.
x ln x
1
e e2
10
1000
1 e
1 1000
131
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- logarithm formulas
- the exponential function sect 7 3 the inverse of the logarithm
- lecture 6 using entropy for evaluating and comparing probability
- zzz 44 44444 sh s1712 w k ss 2
- expand and condense logarithmic expressions worksheet weebly
- chapter 1 functions 1 6 inverse functions and logarithms
- 6 2 properties of logarithms sam houston state university
- chapter 5 exponential logarithmic and inverse trigonometric functions
- frequency domain reverse time migration using the logarithm and the l1
- 3 3 the logarithm as an inverse function sam houston state university
Related searches
- inverse trigonometric ratios calculator
- derivative of inverse trigonometric function
- inverse trigonometric functions
- inverse trigonometric functions worksheet
- inverse trigonometric functions worksheet pdf
- evaluating inverse trigonometric functions practice pdf
- inverse trigonometric ratios worksheet
- inverse trigonometric functions pdf
- inverse trigonometric functions problems
- trigonometric functions and their derivatives
- trigonometric functions problems and answers
- exponential equations and functions worksheet