7.1: Characteristics of Exponential...

Math 12 Pre-Calculus Chapter 7 & 8: Exponential & Logarithmic Functions

1

7.1: Characteristics of Exponential Functions

An exponential function is a function of the form xy c

Ex.1: Graph 2xy using table of values. Then state its domain, range, and asymptote.

Domain: Range: Asymptotes: Y-intercept:

An exponential function is said to be:

Increasing if: c > 1 Decreasing if: 0 < c < 1

Ex.2: Graph the function 1

( )2

x

f x

. Identify the domain, range, intercepts and asymptote.


2

Ex.3: What function of the form xy c can be used to describe the graph shown?

Applications

Exponential functions have a lot of applications to the real world. Generally, they can model

growth (c > 1) or decay (0 < c < 1). A good model for most exponential functions is:

0 ( )t

nA A r where: A = amount at time t

0A = initial amount

r = rate of growth/decay

t = time

n = period of growth

Note: the term half-life refers to the amount of time it takes for decay to be at a rate of 0.5

Ex.4: A strain of bacteria doubles every 5 days. Find the final count of bacteria after 15 days if the

initial population is 200.


3

7.2: Transformations of Exponential Functions

Just as before, we can transform any exponential function as follows:

( )( )b x hy a c k

Ex.1: Consider the base function 3xy . For transformed function 41(3) 5

2

xy

i) state the parameters and describe the corresponding transformations

ii) create a table to show what happens to the given points under each transformation

iii) sketch the graph of the base function and the transformed function

iv) describe the effects on the domain, range, asymptote and intercepts.

Ex.2: A cup of water is heated to 100C and then allowed to cool in a room with an air temperature of

20 C. The temperature, T, in degrees Celsius is measured every minute as a function of time, m, in

minutes. It is found that the temperature of the water decreases exponentially at a rate of 25% every

5 minutes.

a) Draw an approximate graph of this situation.

b) What is the exponential function that describes this situation?

c) Describe the parameters of the transformed graph and how they relate to this situation.


4

7.3: Solving Exponential Equations

When solving exponential equations in Chapter 7 we will have two possible methods:

Method 1 (exact value): Match the bases of both sides of the equation. Then equate the

exponents.

Method 2 (approx. value): If the bases cannot be matched, we can solve using technology.

Ex.1: Rewrite each expression as a power with a base of 3.

a) 27 b) 29 c) 1

23327 81

Matching the Base:

Ex.2: Solve each equation.

a) 2 13 27x x b) 3 2

3116

8

x

x

c) 3 425 1x d) 3 4 255 64

4

y x


5

Graphically:

Ex.3: Solve each equation graphically. Express your answer accurate to 3 significant figures.

a) 4 52 3 6 7x b) 9 5 311 7x x

Application (Compound Interest): 0 1

ntr

A An

Ex.4: You are buying a car! You have saved $5000, but the car costs $5900. You invest your money

at 6.12%, compounded quarterly – how long will it take until you have enough?


6

8.1: Understanding Logarithms

Logarithm (log) is just another representation for the value of an exponent. It is the power to

which the base must be raised to produce a given number. Simply, logarithms are a way of

describing the power of a term. It represents exponent. For example, log10 100 means:

base (10) to the power of what value would result in (100) ?

As base 10 is the most common logarithm in science, it is written as log without the 10.

Ex.1: Using a calculator, evaluate:

a) log 100 b) log 10 c) log 1

d) log 0 e) log 0.1 f) log (-10)

Which of the above was ‘undefined’? Why might this be?

We can exchange between exponential and logarithmic forms as:

logb x a ↔ ab x

Ex.2: Rewrite each of the following from logarithmic form to exponential form:

a) log100 2 b) log50 y

c) 7log 2a d) 2log 1 3x x

Ex.3: Rewrite each of the following from exponential form to logarithmic form:

a) 52 32 b) 410 0.0001

c) 3 4x d) 310x y


7

Ex.4: Find each exact value by re-writing in exponential form (no calculators).

a) log100 b) 2log 0.16

c) 3

1log

81 d) log 64 3x

Graph of Logarithms

The graph of a logarithmic function is the inverse of an exponential function.

Ex.5: Graph the function 2logy x

Domain: Range: Asymptote: Y-Intercept:


8

8.2: Transformations of Logarithmic Functions

Just as before, we can transform any logarithmic function as follows:

log ( )cy a b x h k

Ex.1: a) Use transformations to sketch the graph of the function 3log ( 1) 2y x .

b) Identify the following characteristics of the graph of the function:

i) the equation of the asymptote ii) the domain and range

iii) the y-intercept, if it exists iv) the x-intercept, if it exists

Ex.2: a) Use transformations to sketch the graph of the function 2log (2 6)y x

b) Identify the following characteristics of the graph of the function:

i) the equation of the asymptote ii) the domain and range

iii) the y-intercept, if it exists iv) the x-intercept, if it exists


9

Ex.3: The graph shown can be generated by stretching the graph of 4logy x . Write the equation

that describes the graph.

Ex.4: Welders wear helmets fitted with a filter shade to protect from intense light and radiation.

The filter shade number, N, is defined by the function 7( log )

13

Tn

, where T is the

fraction of visible light that passes through the filter. Shade numbers range from 2 to 14

with a lens shade number of 14 allowing the least amount of light. What fraction of light is

passed through a filter of shade number 12?


10

8.3: Laws of Logarithms

Warm-up: Evaluate or solve by re-writing in exponential form.

a) 5log 125 b) 3

4log 4 c) 1

log 92

x

There are several laws that allow us to combine several logarithms into one statement. These laws will become extremely helpful when we start solving equations involving logarithms. Since logarithms are exponents, the laws are all related to the laws of powers.

Product Law log log logc c cMN M N

Quotient Law log log logc c c

MM N

N

Power Law log logP

c cM P M

Ex. 1: Write each expression in terms of individual logarithms of x, y and z.

a) 5log

xy

z

b) 3

7log x

c) 3

logx

y z

Ex. 2: Use the laws of logarithms to simplify into a single expression. Then evaluate.

a) 6 6 6log 8 log 9 log 2 b) 7log 7 7


11

c) 2 2 2

12log 12 log 6 log 27

3

Ex. 3: Write each expression as a single logarithm in simplest form. State any restrictions on the variable.

a) 2 77 7

5loglog log

2

xx x b) 2

5 5log (2 2) log ( 2 3)x x x

Change of Base log

loglog

cb

c

aa

b

Ex. 4: Find each logarithm to 3 decimal points using technology.

a) 3log 7 b) log 7e

Ex.5: The pH scale is used to measure the acidity or alkalinity of a solution. The pH of a solution

is defined as pH = -log[H+], where [H+] is the hydrogen ion concentration in mol/L. A

neutral solution has a pH of 7, acids have pH < 7 and bases have pH > 7.

a) A common ingredient in cola is phosphoric acid. If a cola has a pH of 2.5, how much

more acidic is it than water?

b) An apple is 5 times as acidic as a pear. If a pear has a pH of 3.8, then what is the pH of an

apple?


12

8.4 (Part 1): Solving Logarithmic Equations

Ex.1: State the restrictions on x for each of the following equations:

a) log2 x 2 1 b) log3(x 4) log5(3 x) log2

When solving equations involving logarithms there are two general forms:

1) log logc cL R If you can combine both sides to a logarithm of the same base, then you can

equate the contents of the logarithm!

2) logc L M If you have a logarithm equal to a numeric value, re-write in exponential

form!

Note: Logarithms are not defined for all real numbers, so we must check our solution(s) for extraneous

roots, or state the restrictions.

Ex.2: Solve each of the following and check your solution(s).

a) log2 x log2(x 2) 3 b) log5(2x 1) log5(x 2) 1

c) log(x 6) log(x 2) log5 d) log3(2x 4) log3(x 1) log3 8


13

Ex.3: Solve.

a) 3 3log ( 1) log 0x x b) 2 2(log ) log 3x x

8.4 (Part 2): Solving Logarithmic Equations & Applications

If we are solving an exponential equation that cannot match the base, we can now use logarithms (finally!)

Power rule for logarithms allows us to take the exponent out in front and solve!

Ex.1: Solve for x

a) 7x1 9 b) 3x2 52x3

c) 3 8x2 51x


14

Ex. 2: Strontium-90 has a half-life of 28 years. How long will it take a 85 g sample to decay to 15g?

(Use 0 ( )t

nA A r )

Ex. 3: How long would it take a population of bees to triple if they multiply 8 fold every 5 weeks?

Ex. 4: How much more intense is an earthquake of 9.8 than an earthquake of 5.4 on the Richter Scale?

Ex. 5: If an earthquake registers 2.7, how high on the Richter scale does an earthquake 52 000 times

stronger register at?


15

Natural Logarithm and the number e

Like , the number e is irrational. It is used in growth and decay situations that are continuous, such as

the increase in world population or the decrease in the mass of a snowball as it melts. The number e (to

nine decimals accuracy) can be found on your calculator with the ex

button.

Little known fact: Ms. Dobson once dressed up as this number for a costume party!

You will meet e again in a more meaningful way when you take calculus.

With calculus, mathematicians have developed the following formula for continuous growth or decay:

A A0ekt

Where A0 is the initial amount, k is a constant (usually a growth rate or a decay rate), and A is the new

amount resulting after a certain time t .

Ex.1: The population of a particular country is 25 million. Assuming the population is growing

continuously, the population P , in millions, t years from now can be determined by the formula

P 25e0.022t . What will be the population, in millions, 20 years from now?

Natural Logarithms

Logarithms with the base of e are called natural logarithms. The natural logarithm of a number x is

written as ln x (pronounced “lawn x “).

In other words, loge x ln x .

You should be able to evaluate the following without a calculator:

lne lne4

ln1 eln 202


16

Natural logs obey the same laws and properties as all other logarithms.

ln x ln y ln(xy) ln x ln y lnx

y

ln xn n ln x eln x x

Ex.2: Solve for x to 3 decimal places

a) 73 xe b) 0.0451500 5 xe c) ln3.6 0.034x

Ex.3: In 1997,the population of Calgary was 795 000. This was an increase of 3.06% from the previous

year. Assuming that the population is growing continuously, the population P , in thousands, can be

determined by the formula 0.0306795 tP e where t is the time in years since 1997.

a) What is the population of Calgary in the year 2000?

b) What year would the population exceed one million?

c) If the population grew at 4.01%, how would the formula change?

Ex. 4: Express 7 as a power of e . Ex. 5: Express e as a power of 12.

7.1: Characteristics of Exponential...

Documents

Transcript of 7.1: Characteristics of Exponential...