MATH 108

Sections 4.1-4.2

Exponential Functions

The term 27 is called a power. If a number is in exponential form, the exponent represents how many times the base is to be used as a factor.

7

ExponentBase

2

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RULES OF EXPONENTS

Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

,x y x ya a a

,x

x yy

aa

a

,x x xab a b

,yx xya a

0 1,a

1 1.

xx

xa

a a

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EXAMPLE 6 Sketching Graphs

Use transformations to sketch the graph of each function.

3 4xf x a.

State the domain and range of each function and the horizontal asymptote of its graph.

1 3xf x b.

3xf x c. 3 2xf x d.

Note that the graph of ex is between 2x and 3x, because 2<e<3. In the first quadrant, 2x<ex<3x; in the second quadrant, 3x<ex<2x. All 3 graphs pass through (0,1).

43 -1 2 13

Solve each exponential equation.1

(a) 2 32 (b) x x xx

e ee

Solve:

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EXAMPLE 8 Bacterial Growth

A technician to the French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubles every hour. If the bacteria count B(t) is modeled by the equation

a. the initial number of bacteria,b. the number of bacteria after 10 hours; andc. the time when the number of bacteria will be

32,000.

with t in hours, find

Find the amount A that results from investing a principal P of $2000 at an annual rate r of 12% compounded continuously for a time t of 3 years.