Algebra II Notes Unit Eight: Exponential and Logarithmic Functions

Algebra II Notes – Unit Eight: Exponential and Logarithmic Functions

Syllabus Objective:8.2 – The student will graph logarithmic and exponential functions including base e.

Exponential Function: ,

Graph of an Exponential Function

Ex: Create a table of values to graph the exponential function .

End Behavior: As .

As , therefore is an asymptote.

Explore: Graph the following exponential functions and describe the change from the graph of .

1. General characteristics of :
2. Graph of is shifted horizontally by h units.
3. Graph of is shifted vertically by k units.
4. If a > 0 and b > 1, it is an exponential growth function.
5. Domain of an Exponential Growth Function: All Reals
6. Range of an Exponential Growth Function:

Ex: Sketch the graph of

Step One: Sketch the graph of

Step Two: Sketch the graph of by reflecting over the x-axis.

Step Three: Translate the graph vertically 4 units up.

Exponential Growth Models: In a real-life situation, if a quantity increases by r percent each time period t, the situation can be modeled by the equation , where a is the initial amount. The quantity is called the growth factor.

Ex: In 1990 the cost of tuition at a state university was $4300. During the next 8 years, the tuition rose 4% each year. Write a model that gives the tuition y (in dollars) t years after 1990. Then estimate the cost of tuition in 1999.

Model:

1999:

Compound Interest: A = amount in account after t years, P = principal (amount when t = 0, r = annual interest rate, n = number of times per year the interest is compounded

Ex: Jane deposits $1500 in an account that pays 6% annual interest. Find the balance after 3 years if the interest is compounded semiannually.

P = 1500, r = 0.06, t = 3, n = 2

You Try:1. Sketch the graph of the exponential function.

2. Use the example above and find out the balance of Jane’s account if the interest is compounded quarterly.

QOD:What is the difference between percent increase and growth factor?

Sample CCSD Common Exam Practice Question(s):

What function describes the graph below?

Sample SAT Question(s): Taken from College Board online practice problems.

1. If x is a positive integer, what is one possible value of the units digit of after it has been multiplied out?

Grid-In

Syllabus Objective: 8.2 – The student will graph logarithmic and exponential functions including base e.

Exponential Decay Function: , where a > 0 and 0 < b < 1

Domain: All Real Numbers; Range: y > 0

Ex: Graph the function and state its domain and range.

y – intercept: Asymptote:

Domain: All real numbers

Range:

Ex: State whether is an exponential growth or decay function.

1. DECAY (because and )
2. GROWTH (because and )
3. DECAY (Can be rewritten as .)
4. GROWTH (because and )
5. DECAY (because and )

Exponential Decay Models: In a real-life situation, if a quantity decreases by r percent each time period t, the situation can be modeled by the equation , where a is the initial amount. The quantity is called the decay factor.

Ex: There are 40,000 homes in a certain city. Each year 10% of the homes are expected to disconnect from septic systems and connect to the sewer system. Write an exponential decay model for the number of homes that still use septic systems. Use the graph of the model to estimate when about 17,200 homes will still not be connected to the sewer system.

Model:

Graph: On the graphing calculator, graph and . Find the point of intersection.

There will be 17,200 homes not connected to the sewer system after about 8 years.

You Try: A new car costs $23,000. The value decreases by 15% each year. Write an exponential decay model for the car’s value. Use the model to estimate the value of the car after 3 years.

QOD: Describe the end behaviors of an exponential decay function.

Sample CCSD Common Exam Practice Question(s):

A company commits to reducing its carbon emissions by 10% each year from the preceding year. If the company emitted 20,000 tons of carbon dioxide in the year prior to starting the program, which formula for represents the company’s emissions during year n of the program?

Sample SAT Question(s): Taken from College Board online practice problems.

The function above can be used to model the population of a certain endangered species of animal. If gives the number of the species living tdecades after the year 1900, which of the following is true about the population of the species from 1900 to 1920?

(A)It increased by about 1,000.

(B)It increased by about 320.

(C)It decreased by about 180.

(D)It decreased by about 320.

(E)It decreased by about 1,000.

Syllabus Objective: 8.2 – The student will graph logarithmic and exponential functions including base e.

Exploration: Evaluate the following for larger and larger values of n.

Note: Students should get values closer and closer to 2.71828… , which is the value of e.

Now use the e key on the calculator, and it will automatically give an approximation of e.

Euler Number: the irrational natural base

Note: e is named after its discoverer, Leonhard Euler.

Simplifying Expressions with Base e

Ex: Simplify the expression .

Use the product of powers property.

Ex: Simplify the expression .

Use the power of product and power of power properties.

Use the quotient of powers property.

Use the definition of negative exponents.

Graphing Natural Base Functions

Ex: Graph the functions and . State the domain and range.

Exponential GrowthExponential Decay

Domain: All real numbers. Range: Domain: All real numbers. Range:

Ex: Sketch the graph of the function. State its domain and range.

Graph the function . Because the coefficient of e is −1, we will reflect over the x-axis. Then the graph will be shifted left 2 and up 3.

Domain: All real numbers

Range:

Continuously Compounded Interest

Recall: is the formula for interest compounded n times per year. If interest is compounded continuously, then . Therefore, the formula for continuously compounded interest is .

Ex: Chelli deposited $1500 into an account that pays 7.5% annual interest compounded continuously. What is her balance after 2 years?

Use with , , and .

You Try: The atmospheric pressure P (in pounds per square inch) of an object d miles above sea level can be modeled by . How much pressure per square inch would you experience at the summit of Mount Washington, 6288 feet above sea level? Graph the model and estimate your height above sea level if you experience 13.23 lb/in2 of pressure.

QOD: e is an irrational number. Define irrational and give 3 other examples of irrational numbers.
Syllabus Objective: 8.3 – The student will evaluate and simplify expressions using properties of logarithms.

Definition of a Logarithm: Let b and y be positive numbers, with .

if and only if

Note: Evaluating a logarithm is the same as finding an exponent.

is read “log base b of y.”

Writing Logarithmic Equations in Exponential Form

Ex: Rewrite the following in exponential form.

1. The base is 2, the exponent is 3, and the power is 8.
2. The base is 3, the exponent is −1, and the power is .
3. The base is 10, the exponent is −2, and the power is 0.01

Note: The logarithmic and exponential forms of the equations are equivalent.

Evaluating Logarithms: To evaluate a logarithmic expression, remember you are finding the exponent the base would need to be raised to in order to obtain the argument of the logarithm.

Ex: Evaluate the following expressions.

1. Answer the question: What exponent would I raise 4 to in order to obtain 64?

because

1. Answer the question: What exponent would I raise 1/3 to in order to obtain 9?

because

1. Answer the question: What exponent would I raise 9 to in order to obtain 3?

because

Special Values of Logarithms: (Let b be a positive real number other than 1.)

Answer the question: What exponent would I raise b to in order to obtain 1? Because any positive real number raised to the 0 power is equal to 1, we can say that .

Answer the question: What exponent would I raise b to in order to obtain b? Because any number raised to the power of 1 is equal to itself, we can say that .

Special Logarithms

Common Logarithm: the logarithm with base 10

Natural Logarithm: the logarithm with base e

Ex: Evaluate .

Answer the question: What exponent would I raise 10 to in order to obtain 100?

because

Ex: Evaluate .

Answer the question: What exponent would I raise e to in order to obtain ?

Logarithms on the Calculator: The calculator can only evaluate common and natural logarithms.

Ex: Approximate the value of .

Ex: Approximate the value of .

You Try: Evaluate the following logarithmic expressions.

1. 2. 3. 4.

QOD: Explain why the logarithm of a negative number is undefined.

Syllabus Objective: 8.2 – The student will graph logarithmic and exponential functions includingbase e.

Exponential and logarithmic functions are inverse functions.

Recall: By definition, if and are inverse functions, then

Therefore, and .

Ex: Evaluate the expressions.

1. Rewrite the argument to match the base.

Finding Inverses

Ex: Find the inverse of the function .

The inverse of a logarithmic function is an exponential function.

Ex: Find the inverse of the function .

Step One: Switch the x and the y.

Step Two: Solve for y. Write in exponential form.

Graphing Logarithmic Functions

Recall: Graph the exponential function .

The graph of (the inverse of ) is the reflection of the graph of over the line .

Domain: ; Range: All real numbersAsymptote:

Note: The domain and range of a logarithmic function is the domain and range of its inverse (exponential function) switched!

Ex: Graph the logarithmic function. State the domain and range.

Plot “convenient” points: Let .

Let .

Because the graph is shifted to the left 2, there is an asymptote at .

Domain: ; Range: All real numbers

Ex: Sketch the graph of the logarithmic function. State the domain and range.

Plot “convenient” points: Let .

Let .

Let .

The graph is shifted down 3, and there is an asymptote at .

Domain: ; Range: All real numbers

Ex: The slope s of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach by the equation . Graph the model and estimate the average diameter of the sand pebbles for a beach whose slope is 0.15.

The average diameter is about 0.84 millimeters.

You Try: Sketch the graph of .

QOD: For what values of b does the graph of intersect its inverse ?

Sample CCSD Common Exam Practice Question(s):

What graph represents the function ?

Syllabus Objectives: 8.1 – The student will simplify an expression involving real-number exponents using laws of exponents. 8.3 – The student will evaluate and simplify expressions using properties of logarithms.

Review: Properties of Exponents (Allow students to come up with these on their own.) We will now extend these properties for use with logarithms.

Let a and b be real numbers, and let m and n be integers.

Product of Powers Property

Quotient of Powers Property

Power of a Power Property

Because of the relationship between logarithms and exponents, the properties of logarithms are similar.

Properties of Logarithms: Let b, r, and v be positive numbers with .

Product Property

Quotient Property

Power Property

Using Properties of Logarithms to Approximate the Value of a Logarithmic Expression

Ex: Use the approximations and to approximate the expressions.

1. Rewrite as a product:

Use the product property of logarithms.

Substitute the values of the logarithms.

1. Use the property of logarithms.

Substitute the values of the logarithms.

1. Rewrite as a power:

Use the power property of logarithms.

Substitute the values of the logarithms.

Rewriting Logarithmic Expressions

Ex: Expand the expression . Assume all variables are positive.

Quotient Property:

Product Property:

Power Property:

Ex: Write an equivalent form of the expression . Assume all variables are positive.

Power Property:

Quotient Property:

Evaluating Logarithms of Base b

Start with .

Rewrite in exponential form:

Take the “log” of both sides (Any log will do!): We will use either the common or natural logarithm because these are the logs that the calculator can evaluate.

Use the power property:

Solve for x:

Change of Base Formula:

Ex: Use the change of base formula to approximate the value of .

Using common logarithms:

Now try the same example using natural logarithms:

Note: We approximated the value of this logarithm previously in the notes. Compare!

Graphing Logarithmic Functions: We can now use the change of base formula to graph logarithmic functions of any base on the graphing calculator.

Ex: Graph the function .

Rewrite the function using the change of base formula. (We will use the natural logarithm.)

Note: We graphed this function by hand earlier. Compare!

You Try: Expand the expression . Assume all variables are positive.

QOD: Explain how you can rewrite the expression without using the quotient property of logarithms. (Hint: Use the power and product properties.)

Sample CCSD Common Exam Practice Question(s):

Expand the expression .

Syllabus Objective: 8.4 – The student will solve exponential and logarithmic equations including base e.

Solving Exponential Equations

Method 1: Rewrite both sides of the equation so that they have the same base.

Note: If then .

Ex: Solve the equation .

Rewrite both sides with a base of 2. (Note: and )

Use the power of a power property.

Equate the exponents and solve for x.

Check the solution by substituting into the original equation. True

Method 2: Taking a logarithm of both sides.

Ex: Solve the equation .

Take the log base 3 of both sides.

Simplify using inverses.

Use the change of base formula to evaluate.

Alternate Method:

Take the natural log of both sides.

Use the power property of logs.

Solve for x.

Check your answer using the calculator.

Ex: Solve the equation .

Isolate the exponential term.

Take the common logarithm of both sides.

Use the power property of logs.

Solve for x.

Check the solution.

Because of the complexity of the solution, a good way to check would be to graph both sides of the original equation on the graphing calculator and find the point of intersection.

Solving a Logarithmic Equation

Method 1: If the logarithms on both sides have the same base, use the fact that if and only if .

Ex: Solve the equation .

Both sides have a logarithm with base 4, so we can equate the arguments.

Solve for x.

Check the solution in the original equation.

Method 2: Rewriting the equation in exponential form.

Ex: Solve the equation .

Rewrite in exponential form.

Solve for x.

Check the solution in the original equation. True

Simplifying Before Solving in a Logarithmic Equation

Ex: Solve the equation .

Rewrite the left side of the equation using the product property of logs.

Write the equation in exponential form.

Solve for x.

Check the solutions in the original equation.

:Not possible to take the log of a negative number!

: True

Solution: (Note: −9 is an extraneous solution)

You Try: Solve the equations.

QOD: Explain why logarithmic equations can have extraneous solutions.

Sample CCSD Common Exam Practice Question(s):

1. Which equation is equivalent to ?

1. What is the solution of the equation ?
2. Solve the equation for n.
3. What is the value of x if ?
4. 1
5. 3

Syllabus Objective: 8.5 – The student will develop mathematical models using exponential or logarithmic equations to solve real world problems.

Writing an Exponential Function: two points determine a unique exponential function

Ex: Write an exponential function whose graph passes through and .

Substitute each ordered pair in for x and y in the equation :

To eliminate a, divide the two equations. Put the highest power of b on top:

Solve for b: (Note: In an exponential function, b cannot be negative.)

Substitute this value of b into one of the original equations and solve for a:

Exponential Function:

Writing a Power Function: two points determine a unique power function

Ex: Write a power function whose graph passes through and .

Substitute each ordered pair in for x and y in the equation :

Solve one of the equations for a:

Substitute the expression into the other equation for a:

Use the power of a quotient property to simplify:

Solve for b using logarithms:

Substitute this value in for b to solve for a:

Power Function:

Using Exponential and Power Models

Ex: Find an exponential model to fit the data. Use the model to estimate y when x is 15.

Enter the x-values into L1 and the y-values into L2 (Go to STAT – Edit)

On the Home screen, go to STAT – CALC and choose option 0, ExpReg. Then type Y1 (found in the VARS menu), and press Enter. This will calculate the exponential regression model and store it in Y1.

Graph the scatter plot along with the exponential regression equation to see if the model fits the data. Use ZoomStat.

Exponential Model: When x is 15,

Ex: The ordered pairs describe the circular area r (square feet) that oil from a leaking oil tanker covers t minutes after it begins leaking. Find a power model for the data. Use the model to estimate the area that will be covered by the leaking oil after 2 hours.

Enter the t-values into L1 and the r-values into L2 (Go to STAT – Edit)

On the Home screen, go to STAT – CALC and choose option A, PwrReg. Then type Y1 (found in the VARS menu), and press Enter. This will calculate the power regression model and store it in Y1.

Graph the scatter plot along with the exponential regression equation to see if the model fits the data. Use ZoomStat.

Power Model: