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5.1 Exponential Functions

Essential Question(s):

• How do you graph exponential functions?

• How do you solve problems involving compound interest?

Exponential Function [pic] where b > 0 and b[pic]1

| | | |

| |Domain |Real Numbers |

| | |[pic] |

| | | |

|[pic] | | |

| | | |

| |Range |Positive Real Numbers |

| | |[pic] |

| | |[pic] |

| |y-intercept | |

| | |The x-axis |

| |Asymptote | |

| | |increasing |

| |If b > 1 | |

| | |decreasing |

| |If 0 < b < 1 | |

Examples:

| | | | | | | |

|Graph [pic] |x |y | |Graph [pic] |x |y |

| |

|[pic] |

| |

|[pic] is a reflection of [pic]across the y-axis |

The natural base e e = 2.718281828459

Note: e is the limit of the expression of [pic] as x gets larger and larger.

The natural exponential function ( [pic]

| | | |

|Sketch the graph of [pic]and [pic]. | | |

| | | |

| |x |y |

| | | |

|[pic] |-2 |0.135 |

| |-1 |0.368 |

| |0 |1 |

| |1 |2.718 |

| |2 |7.389 |

| | |

|Transformation |Equation ( effect on graph of [pic] |

|Vertical translation | |

| |[pic] ( upward shift |

| | |

| |[pic] ( downward shift |

|Horizontal translation | |

| |[pic] ( shift to the left |

| | |

| |[pic] ( shift to the right |

|Reflection | |

| |[pic]( reflection about x-axis |

| | |

| |[pic] ( reflection about y-axis |

|Vertical stretching or shrinking | |

| |[pic] ( stretch if c > 1, shrink if 0 < c < 1 |

|Horizontal stretching or shrinking | |

| |[pic]( shrink if c > 1, stretch if 0 < c < 1 |

For a and b positive, [pic], and m and n real:

1. Exponent laws:

[pic]

[pic]

[pic]

[pic]

[pic]

2. [pic]

3. [pic]

Examples:

Simplify.

1. [pic] 2. [pic] 3. [pic]

[pic] [pic] [pic]

Solve for x.

4. [pic] 5. [pic] 6. [pic]

[pic] [pic] [pic]

Compound Interest – interest computed on your original investment (principal) as well as on any accumulated interest

Compounded semiannually ( compound interest is paid twice a year

Compounded quarterly ( compound interest is paid four times a year

Compounded continuously ( always compounding

Compound interest formula: [pic], where

| | |

|[pic] |Value of account after t years |

| | |

|[pic] |Principal or amount invested |

| | |

|n = |Number of times compounded in one year |

| | |

|r = |Percent in decimal form |

| | |

|t = |Time in years |

Continuous compounding formula: [pic]

Example.

How much interest is earned on a $4000 investment a)compounded semiannually @ 4.5% for 24 months? b) compounded continuously @ 4.5% for 24 months?

| | | | |

|[pic] |? |Compounded semiannually |Compounded continuously |

| | | | |

|[pic] |4000 |[pic] | |

| | | |[pic] |

| | | | |

|n = |2 | | |

| | | | |

|r = |0.045 | | |

| | | | |

|t = |2 | | |

5.2 Exponential Models

Essential Question(s):

• How do you solve problems involving mathematical modeling?

• How do you find the regression equation for data?

| | |

|Doubling time |The time it takes for a population to double |

|Doubling time growth model | |

| |[pic] |

| | |

| | |

| |Alternative formula: |

| | |

| |[pic] |

| | |

| | |

|Half-life | |

| |The time it takes for half of a particular material to decay |

|Half-life decay model | |

| |[pic] |

| | |

| |Alternative formula: |

| |[pic] |

| | |

| | |

|Limited Growth Model |[pic] |

|Logistic Growth Model |[pic] |

|Exponential Regression | |

| |Used to fit a function of the form [pic] to a set of data points. |

|Logistic Regression |Used to fit a function of the form [pic] to a set of data points |

1. The bacteria in a certain culture double every 7.8 hours. The culture has 7,000 bacteria at the start.

a. Write an equation that gives the number of bacteria N in the culture after t hours.

[pic]

b. How many bacteria will the culture contain after 4 hours?

[pic]

2. The radioactive element americium-241 has a half-life of 432 years. Suppose we start with a 20-g mass of americium-241.

a. How much will be left after 200 years? Compute the answer to three significant digits.

[pic]

b. How much will be left after 490 years? Compute the answer to three significant digits.

[pic]

3. An employee is hired to assemble toys. The learning curve[pic] gives the number of toys the average employee is able to assemble per day after t days on the job.

a. How many toys can the average employee assemble per day after 6 days of training? Round to the nearest integer.

[pic]

b. How many toys can the average employee assemble per day after 18 days of training? Round to the nearest integer.

[pic]

c. Does N approach a limiting value as t increases without bound? Explain.

[pic]

Yes, 60 is the upper limit for the number of toys an employee can assemble per day.

Use the data for Radioactive Studies to answer questions 4 - 5:

4. Find an exponential regression model of the form y = abx for the data set.

[pic]

5. Estimate the amount of material remaining after 7 hours. Round to four decimal places.

[pic]

Use the data in the table to the right to answer questions 6 - 8:

|x |y |

|0 |8 |

|10 |20 |

|20 |54 |

|30 |104 |

|40 |191 |

|50 |359 |

|60 |412 |

|70 |429 |

6. Find a logistic regression model [pic] for the data.

[pic]

7. Use the model to find the approximate value of y when x = 53.

[pic]

8. Using the model, what is the projected value of y when x = 70? Why does this differ from the value in the table?

[pic]

The model is used to approximate the data.

5.3 Logarithmic Functions

Essential Question(s):

• How do you graph logarithmic functions?

• How do you change from logarithmic form to exponential form and vice versa?

• What are the properties of logarithmic functions?

Logarithmic functions are the inverse of exponential functions

A. Logarithmic function: [pic] (for x > 0, b > 0 and b [pic]1)

[pic] means…b raised to what power y equals x?

In other words, [pic] (logarithmic form) is equivalent to [pic] (exponential form)

| | | | |

|[pic] |means … |2 raised to what power equals 32? |[pic] |

| | | | |

|[pic] |means… |4 raised to what power equals 2? |[pic] |

| | | | |

|[pic] |means … |b raised to what power equals 1? |[pic] |

| | | | |

|[pic] |means … |b raised to what power equals x? |[pic] |

Examples

| | |Exponential |Logarithmic |

| | | | |

| | | | |

|[pic] | | | |

| |Equation |[pic] |[pic] |

| | |Reals |Positive Reals |

| |Domain | | |

| | |Positive Reals |Reals |

| |Range | | |

| | |[pic] |[pic] |

| |intercept | | |

| | |x-axis |y-axis |

| |Asymptote | | |

| | |increasing | |

| |If b > 1 | |increasing |

| | |decreasing | |

| |If 0 < b < 1 | |decreasing |

B. Natural Logarithm [pic]

Examples

| | |

|[pic] |[pic] |

| | |

|[pic] |[pic] |

C. Properties

| | |Common Logarithm |Natural Logarithm |

| |General |means … |means … |

| | |[pic] or log on calculator |[pic] or ln on calculator |

| | | | |

|1. |[pic] |[pic] |[pic] |

| | | | |

|2. |[pic] |[pic] |[pic] |

| | | | |

|3. |[pic] |[pic] |[pic] |

| | | | |

|4. |[pic] |[pic] |[pic] |

Examples: Evaluate

| |( |[pic] | |[pic] |( | |

|[pic] | | | | | |[pic] |

| |( |[pic] | |[pic] |( |[pic] |

|[pic] | | | | | | |

| |( |[pic] | |[pic] |( |[pic] |

|[pic] | | | | | | |

| |( |[pic] | | | | |

|[pic] | | | | | | |

The domain of [pic] consists of all x for which [pic]. Find the domains of the following logarithmic functions:

| |[pic] | |[pic] |[pic] |

|[pic] |Domain: [pic] | | |Domain: [pic] |

Logarithms are another way of expressing exponents…therefore, connect the laws of exponents with the laws of logarithms.

| | | |

|Product |[pic][pic] |When we multiply two powers with the same base |

|Rule of Logarithms | |we… |

| | | |

| | |ADD EXPONENTS |

| | | |

|Quotient Rule of |[pic][pic] |When we divide powers with the same base we… |

|Logarithms | | |

| | |SUBTRACT EXPONENTS |

| | | |

|Log of a power |[pic][pic] |When we have a power raised to a power we… |

| | | |

| | |MULTIPLY EXPONENTS |

|Expanding Logarithmic Expressions – “break apart” as much as |Condensing Logarithmic Expressions – write as a single logarithmic|

|possible |expression |

|Example |Example |

|Expand [pic] |Condense [pic] |

|[pic] |[pic] |

Change of base formula

[pic]

Example

Simplify [pic]

[pic] OR [pic]

5.4 Logarithmic Models

Essential Question(s):

• How do you find the regression equation for data?

|Decibel | |

| |[pic] |

| | |

| |where |

| |D is the decibel level of the sound |

| |I is the intensity of the sound measured in watts per square meter |

| |[pic] is the intensity of the least audible sound that an average healthy |

| |young person can hear (standardized to be [pic] watts per square |

| |meter). |

|Magnitude M on the Richter scale | |

| |[pic] |

| | |

| |where |

| |E is the energy released by the earthquake (measured in joules) |

| |[pic] is the energy released by a very small reference earthquake |

| |(standardized to be [pic] |

|Velocity of a rocket at burnout | |

| |[pic] |

| | |

| |where |

| |c is the exhaust velocity of the rocket engine |

| |[pic] is the takeoff weight |

| |[pic] is the burnout weight |

|pH scale | |

| |[pic] |

| | |

| |where |

| |[pic] is the hydrogen ion concentration in moles per liter |

| |pH < 7 ( acidic |

| |pH > 7 ( basic |

|Logarithmic regression | |

| |Used to fit a function of the form [pic] to a set of data points. |

Examples

|1. |A rock concert has a volume with an intensity of I = 1.0 × 10–1 W/m2. Find its rating in decibels. |

[pic]

|2. |An earthquake has an energy release of 6.83 × 108 joules. What was its magnitude on the Richter scale? |

[pic]

|3. |A rocket has a weight ratio Wt/Wb = 18.1 and an exhaust velocity c = 4.71 kilometers per second. What is its velocity at |

| |burnout? Compute the answer to two decimal places. |

[pic]

Use the following to answer questions 4-5:

A solution has a hydrogen ion concentration of [H+] = 2.5 × 10–5.

|4. |Find the pH of the solution. Round your answer to one decimal place. |

[pic]

|5. |Is the solution acidic or basic? |

acidic

Use the following to answer questions 6-8:

The following table shows the number of pounds a person lost since beginning a diet.

|Time |Pounds Lost |

|(days) | |

|7 |2 |

|14 |12 |

|21 |17 |

|28 |20 |

|35 |22.5 |

|42 |24 |

|49 |25 |

|56 |25.5 |

|6. |Find a logarithmic regression model for the data. |

[pic]

|7. |Use the regression model to estimate the person's total weight loss after 17 days. |

[pic]

|8. |According to the regression model, what is the projected weight loss for 67 days? |

[pic]

5.5 Exponential and Logarithmic Equations

Essential Question(s):

• How do you solve exponential and logarithmic equations?

Examples

1. Solve [pic] for x to four decimal places.

[pic]

2. Solve [pic] for x to three decimal places.

[pic]

3. Solve [pic], and check.

[pic]

Check:

[pic] [pic]

x = 20

4. Solve [pic], and check.

[pic]

[pic] [pic]

Check:

[pic] [pic]

x = 1, 100

5. Solve [pic] for [pic].

[pic]

-----------------------

Exponential Functions

[pic]

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ÅÙ9[pic]h!Kìhñ`ý5?EHúÿU[pic]-j

ÅÙ9[pic]h!Kìhñ`ý5?U[pic]V[pic]jh!Kìhñ`ý5?U[pic]h!Kìhñ`ý5?6?h!Kìhñ`ý5?jh

G•hñ`ýEHöÿU[pic]jü>M[pic]hñ`ýCJU[pic]V[pic]aJjhñ`ýU[pic]hñ`ý

hñ`ýCJaJ

h…I®hñ`ýjThe Natural Base e

Transformations of Exponential Functions

Exponential Function Properties

Compound Interest

[pic]

where

P = Population at time t

[pic] = Population at time t = 0

d = doubling time

where

y = Population at time t

c = Population at time 0

k= Relative growth rate

where

A = Amount at time t

[pic]= Amount at time t = 0

h = half-life

where

y = Amount at time t

c = Amount at time 0

k= Relative growth rate

Radioactive Studies

|Time in |Grams of |

|Hours |Material |

|x |y |

|0.1 |2 |

|1 |1.2 |

|2 |0.6 |

|3 |0.3 |

|4 |0.2 |

|5 |0.1 |

|6 |0.06 |

[pic]

Note: Inverse functions interchange x and y (domain & range) and are reflections across the line y = x

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