Math 101 - Student Notes & Assignments Unit 4 - Exponential & Logarithmic Functions Unit Objectives 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Find the composition (f ◦ g)(x) of two given functions. Verify by composition that two given functions are inverses. Find inverses algebraically and graph both f (x) and f −1 (x) on the same coordinate system. Identify the domain and range of a function and its inverse. Restrict the domain of a function to find an inverse function. Analyze and graph exponential functions. Solve exponential equations with the same base analytically and graphically. Solve compound interest and continuously compounded models of exponential functions. Convert exponential statements to logarithmic form. Convert logarithmic statements to exponential form. Use change of base to evaluate logarithms. Use the product, quotient, and power properties to expand/condense logarithmic expressions. Analyze and graph logarithmic functions. Identify the domain of a logarithmic function. Use the definition of a logarithm to solve equations. Find the inverse of an exponential function analytically. Solve exponential equations (unlike bases) analytically and graphically. Solve logarithmic equations analytically and graphically. Use u-substitution to solve exponential equations that are quadratic in form. Solve models of growth and decay with exponential and logarithmic equations.

The following problems are to be submitted in class. Show your work neatly and thoroughly. Answers only will not be accepted. The tentative due dates are shown in your Course Calendar. Use this grid to record any changes that may occur in your particular class. Due Date Section hw9 2.6b 5.1 5.2 5.3 hw10 5.4 5.5 5.6

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Problems 64, 69 56, 70 56, 76 28, 72, 90, 94 46, 66 18, 30, 40, 82 6, 20, 26, 37ac, 40

Section 2.6, 5.1 - Compositions & Inverses I. Compositions (2.6) 1. Forming a composition of functions is another way to create a new function from two other functions. The composition operation uses the output of one function as the input of another. 2. Notation: (f ◦ g)(x) = f (g(x)) 3. Evaluate compositions: symbolically (EX 5) and by reading numerical data (tables) or graphs (a) (f ◦ g)(3) = (b) (g ◦ f )(1) = x f (x)

1 3 5 7 5 7 1 3

x g(x)

1 3 5 7 3 5 7 8

4. Find compositions. f (x) = 3x − 1

EX 6 g(x) = x2 − 4

h(x) = x + 2

p(x) =

√

x+3

(a) Evaluate (f ◦ h)(5); (g ◦ p)(1)

(b) Find (g ◦ h)(x) and (h ◦ g)(x)

(c) Find (f ◦ f )(x)

1 5. Special compositions: Given f (x) = x + 2 and g(x) = 4x − 8, find (f ◦ g)(x) , (g ◦ f )(x). 4 What happens?

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II. Inverse Functions (5.1) 1. One-to-one functions. [Horizontal Line Test] EX 1, 2 Definition: A function f (x) is one-to-one if for every two distinct values a and b in the domain, a 6= b implies f (a) 6= f (b) Graphically, one-to-one functions pass the test.

2. Examples of inverse functions: (a) f (x) = 4x − 8 and 1 g(x) = x + 2 4

(b) f (x) = (x − 3)2 , x ≥ 3; and √ g(x) = x + 3 Domain of f ? Range of f ?

(a) x f (x)

x g(x)

(b) x f (x)

Domain of g? Range of g? x g(x)

3. Characteristics and relationships of inverse functions: • If f (x) is one-to-one, then f −1 exists. • f −1 (x) is the symbol for the inverse of f . It is read as ”

is a point on the graph of f −1 .

• If (a, b) is a point on the graph of f , then • Domain of f = Range of f −1

AND

”

Range of f = Domain of f −1

• f (f −1 (x)) = f −1 (f (x)) = x By definition, the composition of a function and its inverse will always result in the original input. • The graphs of a function f and its inverse, f −1 are symmetric about 4. Find the inverse analytically.

. EX 5, 6

(a) Check to see that the function is one-to-one. If not, make a domain restriction so it is one-to-one. (b) Write in y = format (c) Switch x and y, literally (d) Solve for y (e) Rewrite in f −1 (x) format (f) Sketch both on the same coordinate system

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Some examples: Find the inverse of each of the following functions. Note the domain/range for for both f and f −1 . Then sketch both f and f −1 on the same coordinate system. 1. f (x) = 1 − 3x

2. f (x) =

2x + 1 2

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3. f (x) = x2 + 1 , x ≥ 0

Section 5.2 - Exponential Functions Compare: g(x) = x2 and f (x) = 2x Whether the variable, x, appears as an exponent with a constant base or as a base with a constant exponent makes a big difference. The function g(x) is a and the function . f (x) is a new type of function called Definition: f (x) = ax , where a > 0, a 6= 1

 x 1 1. Graphs of basic exponential functions f (x) = a and f (x) = = a−x a Examples: !  x 1 −1 x x = (2 ) or 2−x f (x) = 2 g(x) ! = 2 x

!

! !

2. Characteristics of the graph of an exponential function. • Continuous & one-to-one • Domain • Range • Increasing (when • Asymptote

!

), decreasing (when

• Key points

5

).

EX 1,2

3. Each of the following is a transformation of the graph of f (x) = 2x . Sketch the graph. (a) f (x) = 2x+3 (b) f (x) = 2x − 1 (c) f (x) = 2−x + 3

4. Solve exponential equations of Type I: ab = ac ⇒ b = c ”If the bases are the same, then the exponents are the same.” Solve for x:  3 1 x−3 1. 2 =8 2. = 8x 4

5. Compound Interest model: A = P (1 + nr )nt A= P = r= n= t=

EX 5, 6 3. 274x = 9x+1

EX 8 Typical compounding periods: yearly (annually) n= semi-annual n= quarterly n= monthly n=

6. Natural Base, e 7. Continuous compounding model: A = P ert

EX 9

8. Example: If $20,000 is invested in an account that pays 3.5% annually, find the amount in the account at the end of 4 years if the investment is compounded (a) quarterly

(b) monthly

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(c) continuously.

Section 5.3 - Logarithms & Their Properties Exponential Fcn f (x) = ax y = ax , a > 0, a 6= 1

Inverse x = ay , a > 0, a 6= 1 y = loga x or f (x) = loga x Logarithmic Fcn

Key Idea: A logarithm is an exponent 1. Common logarithms & Natural logarithms

EX 1, 2, 4

2. Convert from exponential form to logarithmic form and vice versa. Solve equations using the definition. (a) Changing exponential expressions to logarithmic expressions. (a) 34 = 81

(b) 100.3010 = 2

(c) e2 = 7.3891

(b) Changing logarithmic expressions to exponential expressions. 1 (b) log4 (a) log49 7 = 12 = −3 (c) ln 0.38 = −0.9676 64

(c) Finding exact values for logarithmic expressions. 8 (a) log2 32 (b) log 10 5 (c) ln e−2

(d) Use a calculator to find the values. (a) log 532 (b) ln 2

(c) log 0.082

(d) 5−3 =

1 125

(d) logm r = t

(d) log4 1

(d) ln (−4)

(e) Solve the following equations for x by writing in exponential form. 1 (a) log5 125 = x (b) log9 x = (c) log4 (x + 2) = 3 2 7

Properties of Logarithms 1. loga 1 = 0 log 1 = 0 ln 1 = 0 Product Rule:

EX 6, 7

2. loga ak = k

3. aloga k = k, k > 0 2log2 5 = 5 eln 3 = 3

log 10 = 1, log3 36 = 6 ln e = 1 4. loga (xy) = loga x + loga y

In words:

Example: log2 32 = log2 4 + log2 8   x = loga x − loga y Quotient Rule: 5. loga y

In words:

Example:   27 = log3 27 − log3 9 log3 9 Power Rule:

6. loga (x)r = r · loga x

In words:

Example: log2 (4)3 = 3 · log2 4

1. Write as a sum, difference, or product of simpler logarithms. • log2 (6x3 y) √ • ln

c ab

2. Write as a single logarithm. • 3 log(x) +

1 log(x + 1) 2

• log5 (x2 − 4) − log5 (x − 2) Because the applications are so prevalent, scientific and graphing calculators have buttons designated to make calculations for base 10 and base e. Use Change of Base to calculate logarithms that are not Base 10 or Base e. EX 8 loga x =

logb x logb a

Example: log7 21 can be calculated by either of the ratios: 8

log 21 ln 21 or log 7 ln 7

Section 5.4 - Logarithmic Functions Exponential Fcn f (x) = ax y = ax , a > 0, a 6= 1

Inverse x = ay , a > 0, a 6= 1 y = loga x or f (x) = loga x Logarithmic Fcn

1. Graphs of logarithmic functions → compare to the related exponential function (its inverse) EX 1, 3 f (x) = 2x

g(x) = log2 x

!

2. Characteristics

!

!

• Domain & Range Find the domain of the functions: f (x) = log4 (x − 3) g(x) = log (x2 − 9) • Increasing (when

), decreasing (when

• Asymptote

EX 2 h(x) = log5 x + 1

).

!

• Key points 3. Graphs of logarithmic functions. Sketch the graph of h(x) = log2 (x − 3)

Sketch the graph of f (x) = ln x − 2

4. (Optional) Finding the inverse of an exponential function. Find the inverse of f (x) = −10x + 4 analytically. 9

EX 6

Section 5.5 - Solving Exponential and Logarithmic Equations Type 1 −→ Bases are the same 1. Express both sides with same base 2. ab = ac ↔ b = c set exponents = and solve

Type 3 −→ Logs on one side 1. Use log properties to express as a single logarithm 2. Use the definition of a logarithm to write in exponential form 3. Solve for x. Give the exact and approximate forms (if appropriate) Quadratic in form 1. Express as: a( )2 + b( ) + c = 0 2. Let u = ( ) 3. Solve for u 4. Proceed as in Type 2 5. Check the result.

Type 2 −→ Bases are different 1. Isolate the exponential expression 2. Take the logarithm of both sides (ln or log) 3. Solve for x (may have to use the distributive property) 4. Express the answer in exact & approximate form Type 4 −→ Logs on both sides 1. Use log properties to express each side as a single logarithm 2. Set arguments = each other 3. Solve for x. 4. Check the solution for inclusion in the domain.

Type 2 Examples: (1)

4x = 15

EX 1,2 (2)

1.2(0.9)x = 0.6

?(3)

10

5x+1 = 112x−4

(4) 6e3x − 3 = 6

Type 3-4 Examples: (1)

log6 (2x + 4) = 2

EX 4,5,6 (2)

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

Quadratic in Form: (1)

32x − 4 · (3x ) = 12

(3)

log5 (x + 3) + log5 (x) = log5 4

EX 9 (2)

e2x + 2ex − 15 = 0

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Section 5.6 - Modeling Exponential growth and decay applications can all be modeled in the form: result = initial ·(growth factor)power Pointers: 1. Focus on the structure of the model 2. The placement of values for initial and result can determine growth or decay • When the initial amount is smaller than the result, there is growth. The exponent will be a positive value (increasing function). • When the initial amount is larger than the result, there is decay. The exponent will be a negative value (decreasing function). 3. The most frequently used function input variable is time. • • • •

When will there be $10,000 in the account? How old is the fossil? How long before the drug falls below safe levels in the blood system? In what year will the population be double what it is now? TIme Period Models
r nt n

A

=

P 1+

A(t)

=

P (1 + r)t

f (x)

=

C · ax

growth factor:

1+

Continuous Models (instantaneous growth/decay) A = A(t)


r n n

=

growth factor:

(1 + r)∗∗

initial value:

result:

(base e) P ert A0 ekt

er ek

a P C

initial value:

A A(t) f (x)

result:

∗∗

P A0 A A(t)

If a population is growing at a steady rate of 2.5%, the yearly growth factor (1 + r) = 1 + .025 or 1.025. If there is a steady decreasing rate of 4%, the growth factor is 1 + (-.04) or .96. Note: In the third time period model above, a represents the yearly growth factor, so a = (1 + r)

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 r nt Compound Interest: A = P 1 + and A = P ert n 1. Suppose you have $3000 to invest. Which investment yields the greater return over 10 years: 6.5% compounded semiannually or 6% compounded continuously? How much more (to the nearest dollar) is yielded by the better investment?

2. How long, to the nearest tenth of a year, will it take $4000 to grow to $6978.43 at 5% annual interest compounded quarterly?

3. Jim invests $15000 in an account paying 3.25% interest. How long will it take to double his investment if the interests is compounded semiannually? continuously?

Instantaneous (continuous, uninhibited) Growth & Decay A(t) = A0 ekt 1. In 1970, the U.S. population was 203.3 million. By 2007, it had grown to 300.9 million. Assume the population is changing exponentially. (a) What is the rate of growth for the population? (b) Find an exponential growth function that models the data for 1970 through 2007. (c) By which year will the U.S. population reach 315 million?

2. A termite colony had a population of 35 on January 1. Three days later, the population was 50. Find the growth rate. If the growth of the colony obeys the continuous model (base e) what will the population be in two weeks?

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Exponential decay models are used in determining the age of fossils and artifacts. The method is based on considering the percentage of carbon-14 remaining in the fossil or artifact. Carbon-14 decays exponentially with a half-life of approximately 5700 years. The half-life of a substance is the time required for half of a given sample to disintegrate. Thus, after 5700 years a given amount of carbon-14 will have decayed to half the original amount. Carbon dating is useful for artifacts or fossils up to 80,000 years old. Older objects do not have enough carbon-14 left to determine age accurately. 1. Use the fact that after 5700 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14. 2. In 1948, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the dead sea scrolls.

Growth & Decay using the model: f (x) = C · ax In this model (compare to the definition of an exponential function in Section 5.2), when a > 1 the function describes growth and when 0 < a < 1 the function describes decay. 1. The function f (x) = 20(.975)x models the percentage of surface sunlight, f (x), that reaches a depth of x feet beneath the surface of the ocean. (a) Is this function increasing or decreasing? How do you know? (b) Determine at what depth, to the nearest foot, there is 1% of surface sunlight.

2. On January 1, 2007, the population of the world was approximately 6.6 billion and was increasing by 1.36% every year. Assume this rate of increase continues. The function f (x) = 6.6 · (1.0136)x , where x is the number of years after 2007, models this situation. What does the model predict the population to be in 2015? In what year will the population have doubled?

3. In 2007 the population of Pakistan was 164 million and it is expected to be 250 million in 2025. (a) Approximate C and a so that P (x) = C · ax−2007 models these data, where P is in millions and x is the year. (b) Estimate when the population will reach 212 million.

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