ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-1 Exponential Growth and Decay

Objectives Write and evaluate exponential expressions to model growth and decay situations. CC.9-12.F.IF.7e;

CC.9-12.F.IF.5; CC.9-12.A.SSE.1

Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an ______________________________________________.

The parent ________________________________ function is f(x) = bx, where the ______________is a

constant and the exponent _______is the independent variable.

The graph of the parent function f(x) = 2x

is shown. The domain is:

The range is:

An ______________________________________ is a line that a graphed function approaches as the value of x gets very large or very small.

A function of the form f(x) = abx

, with a > 0 and b > 1, is an ____________________________ function, which increases as x increases.

When 0 < b < 1, the function is called an _______________________________ function, which decreases as x increases. Tell whether the functions shows growth or decay:

3

104

x

f x

g(x) = 100(1.05)x

You can model growth or decay by a constant percent increase or decrease with the following formula:

In the formula, the base of the exponential expression, 1 + r, is called the __________________________ Similarly, 1 – r is the ______________________________

Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000?

Use a graphics calculator to graph and solve: In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000

A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 8000.

A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100.

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-2 Inverses

Objectives Graph and recognize inverses of relations and functions. CC.9-12.F.FI.5

Find inverses of functions. CC.9-12.F.BF.4c You have seen the word _______________________used in various ways. The additive inverse of 3 is ____. The multiplicative inverse of 5 is _____. You can also find and apply inverses to relations and functions. To graph the _____________________________you can reflect each point across the line _______________. This is equivalent to _______________________the x- and y-values in each ordered pair of the relation. Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation.

When the relation is also a function, you can write the inverse of the function f(x) as ______ This notation does _________indicate a reciprocal. Functions that undo each other are ____________________________ How to find an inverse of a function: The 2 Step: Relationships between a function and its inverse: Use inverse operations to write the inverse of f(x) = 3(x – 7).

x 1 3 4 5 6

y 0 1 2 3 5

Graph f(x) = (2/3)x + 2. Then write the inverse and graph.

Use inverse operations to write the inverse of f(x) = 5x – 7. Then graph both.

A thermometer gives a reading of 25° C. Use the formula C = 5/9(F – 32). Write the inverse function and use it to find the equivalent temperature in °F.

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-3 Converting Exponentials and Logarithms

Objectives

Write equivalent forms for exponential and logarithmic functions. CC.9-12.F.FI.5

Write, evaluate, and graph logarithmic functions. CC.9-12.F.FI.7e How many times would you have to double $1 before you had $8? How many times would you have to double $1 before you had $512? This operation is called finding the______________________. A _____________________________is the _____________________________to which a specified base is raised to obtain a given value. You can write an exponential equation as a logarithmic equation and vice versa.

Write each exponential equation in logarithmic form.

Exponential Equation

Logarithmic Form

35 = 243

25 = 5

104 = 10,000

6–1 =

ab = c

Write each logarithmic form in exponential equation.

Logarithmic Form Exponential Equation

log99 = 1

log2512 = 9

log82 =

log4 = –2

logb1 = 0

A logarithm with base 10 is called a________________________________________. If no base is written for a logarithm, the base is assumed to be_______. log 0.01 = _______ log 0.00001 = ______ log5 125 = _______ log250.04 = ______

Because logarithms are the _______________________of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = ____________.

Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe the domain and range of the inverse function.

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-4 Laws of Logarithms

Objectives Use properties to simplify logarithmic expressions. CC.9-12.A.CED.2; CC.9-12.A.CED.3

Translate between logarithms in any base. CC.9-12.F.FI.7e

Remember that to ________________powers with the same base, you ________ exponents.

With Logarithms: Express log64 + log69 as a single logarithm.

Remember that to ___________powers with the same base, you ______________exponents

With Logarithms: Express these as a single logarithm: log5100 – log54 log749 – log77

Simplify each expression. a. log3311 b. 5log510 c.log100.9 Most calculators calculate logarithms only in base ____ or base ____. You can change a logarithm in one base to a logarithm in another base with the following formula.

Evaluate log328. Evaluate log927. Exponents and Logarithms: The Rule: Examples:

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-5 Solving Exponential and Log Equations

Objectives Solve exponential and logarithmic equations and equalities. CC.9-12.F.LE.4

Solve problems involving exponential and logarithmic equations. CC.9-12.CED.1

An _______________________________________is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations:

Two ways to solve Exponential and Logarithmic Equations: • Try writing them so that the bases are all the same.

Solve and check by getting the same bases first. 32x = 27 98 – x = 27x – 3

• Take the logarithm of both sides.

Solve and check by taking the log of both sides 4x – 1 = 5 23x = 15 A ___________________________is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.

Exponentiating: log6(2x – 1) = –1 log4100 – log4(x + 1) = 1 log5x 4 = 8 Solve: 3 = log 8 + 3log x 2log x – log 4 = 0 2x = 4x – 1 log x2 = 6.

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-6 The Exponential e and Natural Logs

Objectives Use the number e to write and graph exponential functions representing real-world situations. CC.9-

12.F.LE.4; CC.9-12.F.IF.7e; Solve equations and problems involving e or natural logarithms. CC.9-12.A.CED.2

Exponential functions with ____. e is approximately _______. The graph of ____________ The domain: The range

A logarithm with a base of e is called a ______________ logarithm and is abbreviated as ___ The natural logarithmic function: _____________ Since e and ln are inverses, then: Simplify: A. ln e0.15t B. e3ln(x +1) C. ln e2x + ln ex a. ln e3.2 b. e2lnx c. ln ex +4y The formula for _________________________compounded interest is: Where ___is the total amount, ____is the principal, ___is the annual interest rate, and ___ is the time in years. Example: What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously?

What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously?

Radioactive Substances The _________________________of a substance is the time it takes for half of the substance to breakdown or convert to another substance during the process of_________________. Natural decay is modeled by the function below.

Determine how long it will take for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg. Step 1 Find the decay constant for Chromium-51.

Step 2 Write the decay function and solve for t

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-7 Transforming Exponential and Log Equations

Objectives Transform exponential and logarithmic functions by changing parameters. CC.9-12.F.BF.3; CC.9-12.F.FI.5

Describe the effects of changes in the coefficients of exponents and logarithmic functions. CC.9-12.A.CED.3

Graph the following. Describe the asymptote. Tell how the graph is transformed from the graph of the function. f(x) = 2x. g(x) = 2–x + 1 f(x) = 2x – 2

Graph the functions. Find y-intercept and the asymptote. Describe how the graph is transformed from the graph of its parent function.

B. h(x) = e–x + 1

Transforming Logarithmic Equations:

Graph each logarithmic function. Find the asymptote. Describe how the graph is transformed from the graph of its parent function. h(x) = ln(–x + 2) p(x) = –ln(x + 1) – 2.

Write each transformed function. f(x) = 4x is reflected across both axes and moved 2 units down. f(x) = ln x is compressed horizontally by a factor of 1/2 and moved 3 units left. The temperature in oF that milk must be kept at to last n days can be modeled by T(n) = 75 – 16 ln n. Describe how the model is transformed from f(n) = ln n. Use the model to predict how long milk will last if kept at 34oF.

ALGEBRA 2 CH. 7 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 7-8 Modeling data

Objectives Model data by using exponential and logarithmic functions. CC.9-12.A.CED.3

Use exponential and logarithmic models to analyze and predict. CC.9-12.A.CED.2 Analyzing data values can identify a pattern, or repeated relationship, between two quantities. Look at this table of values for the exponential function f(x) = 2(3x).

Notice that the __________of each y-value and the previous one is constant. Each value is _______the one before it, so the ratio of function values is constant for __________spaced x-values. This data can be fit by an exponential function of the form f(x) = abx Determine whether f is an exponential function of x of the form f(x) = abx. If so, find the constant ratio.

x –1 0 1 2 3

f(x) 2 3 5 8 12

Determine whether y is an exponential function of x of the form f(x) = abx. If so, find the equation.

x –1 0 1 2 3

f(x) 2.6 4 6 9 13.5

Using a Graphics Calculator to determine an Equation

You can also use an_____________________________, which is an exponential function that represents a real data set. Once you know that data are exponential, you can use ___________(exponential regression) on your calculator to find a function that fits. This method of using data to find an exponential model is called an____________________.

x –1 0 1 2 3

f(x) 16 24 36 54 81

x –1 0 1 2 3

f(x) –3 2 7 12 17

Find an exponential model for the data. Use the model to predict when the tuition at University of Pewamo will be $6000.

Tuition of the University of Pewamo

Year Tuition

2009-2010 $3128

2010–11 $3585

2011–12 $3776

2012–13 $3950

2013–14 $4188

Use exponential regression to find a function that models this data. When will the number of bacteria reach 2000?

Time (min)

0 1 2 3 4 5

Bacteria 200 248 312 390 489 610

Many natural phenomena can be modeled by natural log functions. You can use a ________________________________________to find a function. Find a natural log model for the data. According to the model, when will the global population exceed 9,000,000,000?

Global Population Growth

Population (billions)

Year

1 1800

2 1927

3 1960

4 1974

5 1987

6 1999