PreCalculus NotesMAT 129

Chapter 6: Exponential and Logarithmic Functions

David J. Gisch

Department of MathematicsDes Moines Area Community College

September 30, 2011

1 Chapter 6Section 6.1: Composite FunctionsSection 6.2: One-to-One Functions; Inverse FunctionsSection 6.3: Exponential FunctionsSection 6.4: Logarithmic FunctionsSection 6.5: Properties of LogarithmsSection 6.6: Logarithmic and Exponential EquationsSection 6.7: Compound InterestSection 6.8: Exponential Growth and Decay; Newton’s Law; EtcSection 6.9: Building Exponential, Logarithmic, and Logistic Modelsfrom Data

Section 6.1: Composite Functions

Summary

We will what composite functions are and how to find theirdomain and range.

Definition

Given two functions f and g , the composite function, denoted by f ◦ g(read as “f composed of g” or “f of g”), is defined by

(f ◦ g)(x) = f (g(x))

The domain of f ◦ g is the set of all numbers x in the domain of g suchthat g(x) is in the domain of f .

Example

Let f (x) = 2x2 − 3 and g(x) = 3x . Find:

(a) (f ◦ g)(1)

(b) (f ◦ g)(3)

(c) (f ◦ g)(−2)

Example

Use the given functions to write one composite function.

(a) Let f (x) = 2x2 − 3 and g(x) = 3x ; (f ◦ g)(x).

(b) Let f (x) = 2x2 − 3 and g(x) = 3x ; (g ◦ f )(x).

(c) Let f (x) = x−4x2+1

and g(x) = 3x ; (f ◦ g)(x).

Finding the Domain of (f ◦ g)(x)

1 You first exclude any elements that are not in the domain of g .

2 You find elements not in the domain of f and set g(x) equal to thosevalues and solve. The resulting solutions are values thet are alsoexcluded from the domain.

Find the Domain

Example

(a) Let f (x) = 2x2 − 3 and g(x) = 3x .Find the domain of (f ◦ g)(x)

(b) Let f (x) = 2x2 − 3 and g(x) = 1x . Find the domain of (f ◦ g)(x)

(c) Let f (x) =√

x − 6 and g(x) = 2x−1 . Find the domain of (f ◦ g)(x)

Example

Given: f (x) =√

x − 4, g(x) = x2 + 1, and h(x) = x−1x+1 , find each of the

following and state the domain.

(a) (f ◦ g)(x)

(b) (h ◦ g)(x)

(c) (g ◦ f )(x)

Section 6.2: One-to-One Functions; InverseFunctions

Summary

We will find out what it means for a function to be one-to-one;find inverse functions by analyzing graphs and equations; andgraph inverse functions.

Definition

A function is one-to-one if for any two inputs in the domain they map totwo different outputs in the range. That is, if x1 6= x2 are in the domainthen f (x1) 6= f (x2).

Theorem

When looking at the graph of a functions f , if every horizontal lineintersects the graph in at most one point then f is one-to-one.

Theorem

If f is one-to-one, then f is said to have an inverse f −1 such that iff (x) = y then f −1(y) = x.

Therefore, if you know that the graph of f (a 1-1 function) contains thepoint (2,−3), then you know f −1 contains the point (−3, 2).

Theorem

The graph of f and f −1 are symmetric with respect to the line y = x.

Example

Given the graph, graph the inverse.

Theorem

Given f and its inverse f −1 then for all values x we have

(f ◦ f −1)(x) = f (f −1(x)) = x and (f −1 ◦ f )(x) = f −1(f (x)) = x

Example

Check if f (x) = 3x − 4 and g(x) = 13x + 4

3 are inverses.

Calculating Inverses

To calculate the inverse of a function you swap the variable y and x andsolve for y .

Example

Find the inverse of f (x) = 2x − 10.

Calculating Inverses

Example

Find the inverse of f (x) = 2x2 − 10.

Calculating Inverses

Example

Find the inverse of f (x) = 2x2 − 10 if x ≥ 0.

Calculating Inverses

Example

Find the inverse of f (x) = 2x+1x+1 .

Section 6.3: Exponential Functions

Summary

We will evaluate exponential functions; graph them; and solveexponential equations.

Theorem

Recall your laws of exponents.

aman = am+n am

an = am−n (an)m = am·n

1m = 1 a0 = 1 a−m = 1am

Definition

An exponential functions is a function of the form

f (x) = ax

where a is a positive real number. The domain of f is the set of all realnumbers.

Theorem

For an exponential function of the form f (x) = ax where a > 0, anda 6= 1, then

f (x + 1)

f (x)= a

Example

Let’s try it for f (x) = 4x .

Graphs of Exponential Functions

http://www.analyzemath.com/expfunction/expfunction.htmlClick here.

f (x) = ax , with a > 1

f (x) = ax , with 0 < a < 1

Example

Graph f (x) = 2−x − 3 using transformations.

The Number e

Definition

The number e is defined as the number such that

e = limn→∞

(1 +

1

n

)nTo see what we mean lets evaluate this for a few values of n.

n 1 2 10 100 100,000 1,000,000(1 + 1

n )n 2 2.25 2.593742 2.704814 2.718268 2.718280

The Graph of e

Figure: Graph of y = ex

Solving Exponential Equations

Example

Solve 22x−1 = 32

Example

Solve e2x−1 = 1e3x·(e−x

)4

Solving Exponential Equations

Example

Section 6.4: Logarithmic Functions

Summary

We will change exponential functions to logarithmic functionsand vice versa; evaluate logarithmic functions; determine thedomaina nd range; and solve logarithmic functions.

Definition

A logarithmic function to the base a, where a > 0 and a 6= 1, is denotedby y = loga x (read as “log base a of x”) and is defined by

y = loga x if and only if x = ay

The domain of a logarithmic function of this form is x > 0.

Example

Change to the alternate form.

(a) For example, 34 = 81, so 4 = log3 81.

(b) y = log3 x

(c) y = log5 x

(d) y = log2 32

Example

Change to the alternate form.

(a) 1.23 = m

(b) eb = 9

(c) −3 = loge x

Domain of a Logarithmic Function

Note:

The reason this is true is because the logarithmic function y = loga x isdefined as the inverse function of y = ax . For any inverse f −1(x) thedomain is the range of f (x). In this case f (x) is an exponential functionwhich has the range of (0,∞).

Domain Range

f (x) = ax (−∞,∞) (0,∞)

f −1(x) = loga x (0,∞) (−∞,∞)

Example

Find the domain of each logarithmic function.

(a) f (x) = log3(x − 2)

(b) g(x) = log2(x+3x−1

)

(c) h(x) = log 12

x2

Graphs of Logarithmic Functions

Recall that if f (x) contains the point (a, b), then f −1(x) contains thepoint (b, a). Thus, as logarithmic functions are inverses of exponentialfunctions we can reflect (about y = x) the graph an exponential functionto create the graph of a logarithmic function.

Graphs of Logarithmic Functions

The Natural Logarithm Function

Rather than write loge we have a special term for this and we call it thenatural logarithm.

y = ln x if and only if x = ey

Example

(a) Find the domain of the logarithmic function f (x) = − ln(x − 3).

(b) Graph f (x).

(c) Determine the range and vertical asymptote.

(d) Find f −1(x).

(e) Use f −1(x) to find the range of f .

(f) graph f −1(x).

Note:

If we have a logarithm of base 10 we call this the common logarithm. Inthis case we do not write the number of the base.

y = log x if and only if x = 10y

Example

(a) Find the domain of the logarithmic function f (x) = 3 log(x + 5).

(b) Graph f (x).

(c) Determine the range and vertical asymptote.

(d) Find f −1(x).

(e) Use f −1(x) to find the range of f .

(f) graph f −1(x).

Solving Logarithmic Equations

Example

Solve each of the following

(a) log3(4x − 7) = 2

(b) log2(2x + 1) = 3

(c) e2x+5 = 8

(d) ln(x + 2) = 5

Example

The formulaD = 5e−0.4h

can be used to find the number of milligrams D of a certain drug that is ina patient’s bloodstream h hours after the drug was administered. Whenthe number of milligrams reaches 2, the drug is to be administered again.What is the time between injections?

Section 6.5: Properties of Logarithms

Summary

We will work with the properties of logarithms to rewritelogarithms in different forms and to evaluate logarithms whosebase is neither 10 nor e.

Properties of Logarithms

loga 1 = 0 (1)

loga a = 1 (2)

aloga M = M (3)

loga ar = r (4)

loga(MN) = loga M + loga N (5)

logaM

N= loga M − loga N (6)

loga M r = r loga M (7)

Example

Write log2(x2 3√

x − 1), with x > 1, as the sum of logarithms.

Example

Write log6x4

(x2+3)2, with x 6= 0, as the difference of logarithms.

Example

Write ln x3√x−2

(x+1)2, with x > 2, as the sum and difference of logarithms.

Example

Write 3 ln 2 + ln(x2) + 2 as a single logarithm.

Example

Write 12 loga 4− 2 loga 5 as a single logarithm.

Change of Base Formula

Theorem

If a 6= 1, b 6= 1 and M are positive real numbers, then

loga M =logb M

logb a

Example

Evaluate log4 9 on your calculator.

Section 6.6: Logarithmic and Exponential Equations

Summary

We will work with the properties of logarithms and exponents tosolve equations.

Example

Solve the following:2 log4 x = log4 9

Example

Solve the following:

log2(x + 2) = log2(1− x) = 1

Example

Solve the following:3x = 7

Example

Solve the following:5 · 2x = 3

Example

Solve the following:2x−1 = 52x+3

Example

Solve the following:5x−2 = 33x+2

Example

Solve the following:4x − 2x − 12 = 0

Example

Solve the following:x + ex = 2

Example

The population of the world in 2006 was 6.53 billion people and wasgrowing at a rate of 1.14% per year. Assuming that this growth ratecontinues we get the equation

P(t) = 6.53(1.0114)t−2006

where t is years since 2006.

(a) According to this model when will the population reach 9.25 billionpeople?

(b) According to this model when will the population reach 11.75 billionpeople?

Section 6.7: Compound Interest

Summary

We will be skipping this section.

Section 6.8: Exponential Growth and Decay;Newton’s Law; Etc

Summary

We will applications of growth and decay with exponentialequations; see how we can apply Newton’s Law; and look atlogistic models.

Uninhibited Growth/Decay

Formula of Growth/Decay

Many natural phenomena have been found to follow the law that anamount A varies with time t according to the function

A(t) = A0ekt

where A0 is the original (initial) amount when t = 0.

If k > 0 then we have growth and if k < 0 we have decay.

Example

A colony of bacteria grows according to the law of uninhibited growthaccording to the function

N(t) = 90e0.05t

where N is measured in grams and t is measure in days.

(a) What is the initial amount of bacteria?

(b) What is the growth rate of the bacteria?

(c) What is the population after 5 days?

(d) How long will it take for the population to reach 140 grams?

(e) What is the doubling time for the population?

Example

A colony of bacteria grows according to the law of uninhibited growth.

(a) If the number of bacteria doubles in 4 hours, find the function thatmodels the cells.

(b) How long will it take for the colony to triple?

(c) How long will it take to double in size again (increase four times)?

Example

Traces of burned wood along with ancient stone tools in an archeologicaldig in Chile were found to contain approximately 1.67% of the originalamount of carbon 14. If the half-life of carbon 14 is 5600 years,approximately when was the tree cut and burned?

Newton’s Law of Cooling

Newton’s Law of Cooling

The temperature u of a heated object at a given time t can be modeled bythe following function:

u(t) = T + (u0 − T )ekt

where T is the constant temperature of the surrounding medium, u0 is theoriginal (initial) temperature, and k is a negative constant.

Example

An object is heated to 100◦C and is then allowed to cool in a room whoseair temperature is 30◦C .

(a) If the temperature of an object is 80◦C after 5 minutes, when will itstemperature be 50◦C ?

(b) How long will it take for the object to reach 35◦C .

Example

A detective is called to the scene of a crime where a dead body has justbeen found. She arrives on the scene at 10:23 pm and begins herinvestigation. Immediately, the temperature of the body is taken and isfound to be 80◦F . The detective checks the programmable thermostat andfinds that the room has been kept at a constant 68◦F for the past 3 days.

After evidence from the crime scene is collected, the temperature of thebody is taken once more and found to be 78.5◦F . This last temperaturereading was taken exactly one hour after the first one. The next day thedetective is asked by another investigator, “What time did our victim die?”

Logistic Model

Logistic Model

In a logistic model, the population P after time t obeys the equation

P(t) =c

1 + ae−bt

where a, b, and c are constants with c > 0. The model is growth if b > 0and decay if b < 0.

The domain is all real numbers, and the range is (0, c).

There are horizontal asymptotes of y = 0 and y = c ; hence the range.

We call c the carrying capacity as it is the maximum that apopulation can grow due to restraints.

Example

Fruit flies are placed in a half-pin milk bottle with a banana (for food) andyeast (for a stimulus to lay eggs). Suppose that the fruit fly populationafter t days is given by

P(t) =230

1 + 56.6e−0.37t

(a) State the carrying capacity and the growth rate.

(b) Determine the initial population.

(c) What is the population in 5 days?

(d) How long will it take for the population to reach 180?

Example

The logistic model

P(t) =0.9

1 + 6e−0.32t

relate the proportion of U.S. households that own a DVD player to theyear. Let t = 0 represent 200, t = 1 represent 2001 and so on.

(a) Determine the maximum proportion of households that will own aDVD player.

(b) What proportion of households owned a DVD player in 2000?

(c) What proportion of households owned a DVD player in 2010?

(d) When will 85% of households own a DVD player?

Section 6.9: Building Exponential, Logarithmic, andLogistic Models from Data

Summary

We will use data to generate equations to predict futureoutcomes. This is another form of regression. Recall that wehave done linear and quadratic regression.

Exponential Regression

Logarithmic Regression

Logistical Regression

Example

Kathleen is interested in finding a function that explains the growth of cellphone usage in the United States. She gathers data on the number (inmillions) of U.S. cell phone subscribers from 1985 through 2005. the datais shown on the next page.

(a) Use the calculator to draw a scatter diagram.

(b) Use the calculator to find an exponential function of best fit(regression equation).

(c) Using this function predict the number of subscribers in 2012.

Example

Example