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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Exponential and Logarithmic Functions9
Exponential and Logarithmic Functions
9.1 Composite and Inverse Functions
9.2 Exponential Functions
9.3 Logarithmic Functions
9.4 Properties of Logarithmic Functions
9.5 Common and Natural Logarithms
9.6 Solving Exponential and Logarithmic Equations
9.7 Applications of Exponential and Logarithmic Functions
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3. Exponential Functions
9.2
Exponential Functions
Graphing Exponential Functions
Equations with x and y Interchanged
Applications of Exponential Functions
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4. Graphing Exponential Functions
In Chapter 7, we studied exponential expressions with rational-number exponents such as 72/3. What about expressions with irrational exponents? To attach meaning to consider a rational approximation, r, of
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5. 1.4 < r < 1.5
1.41 < r < 1.42
1.414 < r < 1.415
Any positive irrational exponent can be interpreted in a similar way. Negative irrational exponents are then defined using reciprocals.
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6. Exponential Function
The function f (x) = ax, where a is a positive constant, is called the exponential function, base a.
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7. Example Solution Graph the exponential function given by 1 –1 2 –2 3 4 3 6 2 5 1 -1 -2 7 8 x y 1 –1 2 –2 3
1/3 9
1/9 27
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8. Example Solution Graph the exponential function given by 4 3 6 2 5 1 -1 -2 7 8 x y 1 –1 2 –2 –3
1/3 3
1/9 9 27
The curve is a reflection of y = 3x.
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9. All graphs of f (x ) = ax go through the y-intercept (0, 1).
From the previous two examples, we can make the following observations.
For a > 1, the graph of f (x ) = ax increases from left to right. The greater the value of a the steeper the curve.
For 0 < a < 1, the graph of f (x ) = ax decreases from left to right. For smaller values of a, the graph becomes steeper.
All graphs of f (x ) = ax go through the y-intercept (0, 1).
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10. All graphs of f (x ) = ax have the x-axis as the asymptote.
If f (x ) = ax, with a > 0, a not 1, the domain of f is all real numbers, and the range of f is all positive real numbers.
For a > 0, a not 1, the function given by f (x ) = ax is one-to-one. Its graph passes the horizontal-line test.
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11. Equations with x and y Interchanged
It will be helpful in later work to be able to graph an equation in which the x and y in y = ax are interchanged.
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12. Example Solution Graph the exponential function given by 1 3 1/3 9 1/9 4 3 6 2 5 1 -1 -2 x y 1 3
1/3 9
1/9 27 –1 2 –2
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13. Applications of Exponential Functions
Example Interest compounded annually.
The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula
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14. Example Solution Suppose that $60,000 is invested at 5% interest,
compounded annually.
a) Find a function for the amount in the account after t years.
b) Find the amount of money in the account at t = 6.
Solution
= $60000( )t
= $60000(1.05)t
b) A(6) = $60000(1.05)6
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