Chapter 3 Exponential & Logarithmic Functions Section 3.1 Exponential Functions & Their Graphs Definition of Exponential Function: The exponential function f with base a is denoted f(x) = ax where a > 0, a≠ 1 and x is any real number. Example 1: Evaluating Exponential Expressions Use a calculator to evaluate each expression a. 2-3.1

b. 2-π

Example 2: Graphs of y = ax In the same coordinate plane, sketch the graph of each function. a. f(x) = 2x

b. g(x) = 4x

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Example 3: Graphs of y = a-x In the same coordinate plane, sketch the graph of each function. a. F(x) = 2-x

b. G(x) = 4-x

Compare the functions in examples 2 and 3. F(x) = 2-x =

and Graph of y = ax

G(x) = 4-x = Graph of y = a-x

Domain Range Intercepts Increasing/ Decreasing Horizontal Asympotote 2

Example 4: Sketching Graphs of Exponential Functions Compare the following graphs to g(x) = 3x a. g(x) = 3x+1

c.

k(x) = -3x

b. h(x) = 3x – 2

d. j(x) = 3-x

The Natural Base e e ≈ 2.71828... is called the natural base. The function f(x) = ex is called the natural exponential function. Example 5: Evaluating the Natural Exponential Function Use a calculator to evaluate each expression. a. e-2

b. e-1

c. e1

d. e2 3

Example 6: Graphing Natural Exponential Functions Sketch the graph of each natural exponential function. a. f(x) = 2e0.24x

b. g(x)= e-0.58x

Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1) For n compoundings per year: A = P(1 +
/)nt 2) For continuous compounding: A = Pert

Example 7: Compounding n Times and Continuously A total of $12,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded a) Quarterly

b) Continuously

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Example 8: Radioactive Decay In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread radioactive chemicals over hundreds of square miles, and the government evacuated the city and the surrounding area. To see why the city is now uninhabited, consider the following model: P = 10e-0.0000845t This model represents the amount of plutonium that remains (from an initial amount of 10 pounds) after t years. Sketch the graph of this function over the interval from t = 0 to t = 100,000. How much of the 10 pounds will remain after 100,000 years?

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Section 3-2: Logarithmic Functions and Their Graphs LOGARITHMIC FUNCTIONS Remember… From Section 1.6 – a function has an inverse if it passes the Horizontal Line Test (no horizontal line intersects the graph more than once). •

Exponential functions of the form

f (x) = a x

pass the Horizontal Line Test, which means they

must have an inverse. •

The inverse of an exponential function is called the logarithmic function with base a.

Try and start finding the inverse of: f ( x) = a

x

Where do you run into trouble?

DEFINITION OF LOGARITHMIC FUNCTION: For

x > 0 and 0 < a ≠1,

y = log a x if and only if x = a y . The function given by

f (x) = log a x is called the logarithmic function with base a.

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Remember… A logarithm is an exponent! This means that be raised to obtain x. For instance,

log a x

is the exponent to which a must

log2 8 = 3 because 2 must be raised to the third power to get 8.

Example 1: Evaluating Logarithms

1 = 100

a. log 2 32 =

d.

log10

b.

log 3 27 =

e.

log 3 1 =

c.

log 4 2 =

f.

log 2 2 =

The logarithmic function with base 10 is called the common logarithmic function. Example #2: Evaluating Logarithms on a Calculator Use a calculator to evaluate each expression. a.

log10 10

b.

2log10 2.5

c.

log10 (−2)

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PROPERTIES OF LOGARITHMS: 1.

log a 1 = 0 because…

2.

log a a = 1 because…

3.

loga a x = x

4. If

because…

loga x = log a y , then x = y .

Example #3: Using Properties of Logarithms Solve each equation for x. List which property you used. a.)

log 2 x = log 2 3

b.)

log4 4 = x

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GRAPHS OF LOGARITHMIC FUNCTIONS Remember… The graphs of inverse functions are reflections of each other in the line

y = x.

Example #4: Graphs of Exponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function. a.

f (x) = 2 x

b.

g(x) = log2 x

Example #5: Sketching the Graph of a Logarithmic Function Sketch the graph of the common logarithmic function

f (x) = log10 x .

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Basic Characteristics of Logarithmic Graphs: Graph of

y = log a x , a > 1

•

Domain:

•

Range:

•

Intercept:

•

Increasing or decreasing?

•

Vertical asymptote:

•

Reflection:

Example #6: Sketching the Graphs of Logarithmic Functions Sketch each of the following, referring to

f (x) = log10 x .

Be sure to describe the

transformation.

a.

g(x) = log10 (x −1)

b.

h(x) = 2 + log10 x

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THE NATURAL LOGARITHMIC FUNCTION The logarithmic function with base e is the natural logarithmic function and is denoted by the special symbol

ln x , read as “el en of x.”

PROPERTIES OF NATURAL LOGARITHMS: 1.

ln1 = 0 because…

2.

lne = 1 because…

3.

ln e x = x

4. If

because…

ln x = ln y , then x = y .

Example #7: Using Properties of Natural Logarithms

1 = a. ln e

b.

lne 2 =

c.

ln e 0 =

d.

2lne =

Example #8: Evaluating the Natural Logarithmic Function Use a calculator to evaluate each expression. a.

ln2

b.

ln0.3

c.

ln e 2

d.

ln( −1)

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Example #9: Finding the Domain of Logarithmic Functions Find the domain of each function. a.

f (x) = ln(x − 2)

b.

g(x) = ln(2 − x)

c.

h(x) = ln x 2

Example #10: Human Memory Model Students participating in a psychological experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model

f (t) = 75 − 6ln(t +1) ,

0 ≤ t ≤ 12

where t is the time in months.

a. What was the average score on the original (t = 0) exam?

b. What was the average score at the end of t = 2 months?

c. What was the average score at the end of t = 6 months? 12

Section 3-3 Properties of Logarithms You know that you can only put ln or log10 into your calculator. In order to evaluate other logs, you will need the change-of-base formula. Change-of-Base Formula Let a, b, and x be positive real numbers such that a≠1 and b≠1. Then loga x is given by loga x =

  

Example 1: Changing Bases Using Common Logarithms a. log430 =

b. log214 =

Example 2: Changing Bases Using Natural Logarithms a. log430 =

b. log214 =

Properties of Logarithms You know from the previous section that the logarithmic function is the inverse of the exponential function. Thus, it makes sense that properties of exponents should have corresponding properties involving logarithms. Properties of Logarithms Let a be a positive number such that a ≠1, and let n be a real number. If u and v are positive real numbers, the following properties are true: 1. logauv = logau + logav 1. ln (uv) = ln u + ln v



2. loga = logau - logav 2. ln  = ln u – ln v 3. logaun = nlogau 3. ln un = n ln u 13

Example 3: Using Properties of Logarithms Write the logarithm in terms of ln 2 and ln 3 a. ln 6

b. ln



Example 4: Using Properties of Logarithms Use the properties of logarithms to verify that – ln ½ = ln 2

Rewriting Logarithmic Expressions Example 5: Rewriting the Logarithm of a Product log105x3y

Example 6: Rewriting the Logarithm of a Quotient ln

√

Example 7: Condensing a Logarithmic Expression  log10x + 3log10(x+1)

Example 8: Condensing a Logarithmic Expression 2ln (x+2) – ln x

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Section 3-4: Exponential and Logarithmic Equations INTRODUCTION In this section, you will study procedures for solving equations involving exponential and logarithmic functions. A simple example:

2 x = 32

This method does not work for an equation as simple as:

ex = 7

The following properties are the INVERSE PROPERTIES of exponential and logarithmic functions.

Base a

Base e

1.

log a a x = x

lne x = x

2.

a log a x = x

e ln x = x

SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS 1. To solve an exponential equation, first isolate the exponential expression, then take the logarithms of both sides and solve for the variable. 2. To solve for a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

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SOLVING EXPONENTIAL EQUATIONS Example 1: Solving an Exponential Equation Solve

e x = 72 .

Example #2: Solving an Exponential Equation Solve

e x + 5 = 60 .

Example #3: Solving an Exponential Equation Solve

4e 2x = 5 .

Example #4: Solving an Exponential Equation Solve

e 2x − 3e x + 2 = 0 .

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SOLVING LOGARITHMIC EQUATIONS To solve a logarithmic equation such as

ln x = 3

Logarithmic form

write the equation in exponential form as follows.

e ln x = e 3

Exponentiate both sides.

x = e3

Exponential form

The procedure is called exponentiating both sides of an equation.

Example #5: Solving a Logarithmic Equation Solve

ln x = 2 .

Example #6: Solving a Logarithmic Equation Solve

5 + 2ln x = 4 .

Example #7: Solving a Logarithmic Equation Solve

2ln 3x = 4.

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Example #8: Solving a Logarithmic Equation Solve

ln x − ln(x −1) = 1.

In solving exponential or logarithmic equations, the following properties are useful. Can you see where these properties were used?

1.

x=y

if and only if

loga x = log a y.

2.

x=y

if and only if

a x = ay , a > 0, a ≠1.

Example #9: Doubling an Investment You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?

Example #10: Consumer Price Index for Sugar From 1970 to 1993, the Consumer Price Index (CPI) value y for a fixed amount of sugar for the year t can be modeled by the equation

y = −169.8 + 86.8ln t where t = 10 represents 1970. During which year did the price of sugar reach 4 times its 1970 price of 30.5 on the CPI?

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Section 3-5: Exponential and Logarithmic Models

INTRODUCTION The five most common types of mathematical models involving exponential and logarithmic functions are as follows: 1. Exponential growth model:

y = ae bx ,

2. Exponential decay model:

y = ae −bx ,

3. Gaussian model: 4. Logistics growth model: 5. Logarithmic models:

y = ae

b>0 b>0

−( x − b )2 c

a y= 1+ be −( x − c ) d y = a + bln x and y = a + blog10 x

The graphs of the basic forms of these functions are as follows:

y = ex

y = e −x

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2 −x y=e

y =1+ ln x

y=

1 1+e−x

y =1+ log x 10

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EXPONENTIAL GROWTH AND DECAY Example 1: Population Increase Estimates of the world population (in millions) from 1980 to 1992 are shown in the table. Year 1980 Population 4453 (in millions)

1985

1986

1987

1988

1989

1990

1991

1992

4850

4936

5024

5112

5202

5294

5384

5478

An exponential growth model that approximates this data is given by

P = 4451e 0.017303 t ,

0 ≤ t ≤ 12

where P is the population (in millions) and t = 0 represents 1980. According to this model, when will the world population reach 6 billion?

Example #2: Finding an Exponential Growth Model Find an exponential growth model whose graph passes through the points (0, 4453) and (7, 5024).

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In living organic material, the ratio of the number of radioactive carbon isotopes (carbon 14) to the number of nonradioactive carbon isotopes (carbon 12) is about 1 to 1012. When organic material dies, its carbon 12 content remains fixed, whereas its radioactive carbon 14 begins to decay with a half-life of about 5700 years. To estimate the age of dead organic material, scientists use the following formula, which denotes the ratio of carbon 14 to carbon 12 present at any time t (in years).

R=

1 −t 8223 12 e 10

Example #3: Carbon Dating The ratio of carbon 14 to carbon 12 in a newly discovered fossil is

R=

1 1013 .

Estimate the age of the fossil.

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GAUSSIAN MODELS Gaussian models are commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve.

Example #4: SAT Scores In 1993, the Scholastic Aptitude Test (SAT) scores for males roughly followed a normal distribution given by

y = 0.0026e −(x − 500)

2

48,000

,

200 ≤ x ≤ 800

where x is the SAT score for mathematics. Sketch the graph of this function. From the graph, estimate the average SAT score.

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LOGISTICS GROWTH MODELS Example #5: Spread of a Virus On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled by

5000 y= −0.8t , 0 ≤ t 1+ 4999e where y is the total number infected after t days. The college will cancel classes when 40% or more of the students are ill. a) How many are infected after 5 days?

b) After how many days will the college cancel classes?

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Example #6: Magnitude of Earthquakes On the Richter scale, the magnitude R of an earthquake of intensity I is given by

I R = log10 I0 where I0 = 1 is the minimum intensity used for comparison. Find the intensities per unit of area for the following earthquakes. (Intensity is a measure of the wave energy of an earthquake.) a) Tokyo and Yokohama, Japan in 1923, R = 8.3.

b) Haiti Region in 2010, R = 7.0.

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