section 9.3 we have previously worked with exponential expressions, where the exponent was a...

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Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined for all real numbers, x, including irrational numbers. However, the proof of this would have to wait until a higher level math course.

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Page 1: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Section 9.3

• We have previously worked with exponential expressions, where the exponent was a rational number

• The expression bx can actually be defined for all real numbers, x, including irrational numbers.

• However, the proof of this would have to wait until a higher level math course.

Page 2: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

A function of the form f(x) = bx is called an exponential function if b > 0, b is not 1, and x is a real number.

Page 3: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

We can graph exponential functions of the form f(x) = 3x, g(x) = 5x or h(x) = (½)x by substituting in values for x, and finding the corresponding function values to get ordered pairs.

We would find all graphs satisfy the following properties:

• 1-to-1 function

• y-intercept (0, 1)

• no x-intercept

• domain is (-, )

• range is (0, )

Page 4: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

We would find a pattern in the graphs of all the exponential functions of the type bx, where b > 1.

x

y

Page 5: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

We would find a pattern in the graphs of all the exponential functions of the type bx, where 0 < b < 1.

x

y

Page 6: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

We would find a pattern in the graphs of all the exponential functions of the type bx-h, where b > 1.

x

y

(h, 1)

The graph has the same shape as the graph for bx, except it is shifted to the right h units.

Page 7: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

We would find a pattern in the graphs of all the exponential functions of the type bx+h, where b > 1.

x

y

(-h, 1)

The graph has the same shape as the graph for bx, except it is shifted to the left h units.

Page 8: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Since an exponential function is a 1-to-1 function,

if b > 0, and b 1, then bx = by is equivalent to x = y.

Example

Solve 6x = 36

6x = 62

x = 2

Page 9: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example

Solve 92x+1 = 81

92x+1 = 92

2x + 1 = 2

2x = 1

x = ½

Page 10: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example

Solve x2327

1

3-3 = 32x

-3 = 2x

2

3x

Page 11: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example

Solve 43x-6 = 322x

(22)3x-6 = (25)2x

(22)3x-6 = 210x

26x-12 = 210x

6x-12 = 10x

-12 = 4x

x = -3

Page 12: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

• Many applications use exponential functions of various types.

• Compound interest formulas are exponential functions used to determine the amount of money accumulated or borrowed.

• Exponential functions with negative exponents can be used to describe situations of decay, while those with positive exponents can be used to describe situations of growth.

Page 13: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example

Find the total amount invested in a savings account if $5000 was invested and earned 6% compounded monthly for 18 years. Round your answer to two decimal places.

The formula that is used for calculating compound interest isnt

n

rPA

1

where P is the initial principal invested, r is the interest rate, n is the number of times interest is compounded each year, t is the time of the investment (in years) and A is the amount of money in the account.

Page 14: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example (cont.)

1812

12

06.0150001

nt

n

rPA

216005.015000

216005.15000

14683.83$

Page 15: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example

An accidental spill of 100 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of 5% each day. Find how much debris still remains after 30 days.

The formula that would be used for this problem is

rtAy )7.2(

where A is the amount of radioactive material to start, r is the rate of decay, t is the number of days and y is the amount of radioactive material after the time period.

Page 16: Section 9.3 We have previously worked with exponential expressions, where the exponent was a rational number The expression b x can actually be defined

Example (cont.)

3005.0)7.2(100)7.2( rtAy5.1)7.2(100 (exact answer)

22.54 grams