Pre-Calculus

1/3/2007

State “exponential growth” or “exponential decay”(no calculator needed)

State “exponential growth” or “exponential decay”(no calculator needed)

a.) y = e2x b.) y = e–2x

c.) y = 2–x d.) y = 0.6–x

k > 0, exponential growth k < 0, exponential decay

b>1 so growth, but reflect over y-axis, so decay

0<b<1 so decay, but reflect over y-axis, so growth

Pre-Calculus

1/3/2007

Characteristics of a Basic Exponential Function: Characteristics of a Basic Exponential Function:

Domain:

Range:

Continuity:

Symmetry:

Boundedness:

Extrema:

Asymptotes:

End Behavior:

( - , )

( 0, )

continuous

none

b = 0

none

y = 0

lim f(x) x =

lim f(x) x - = 0

Pre-Calculus

1/3/2007

Pre-Calculus

1/3/2007

QuestionQuestion

Use properties of logarithms to rewrite the expression as a single logarithm.

log x + log y 1/5 log z

log x + log 5 2 ln x + 3 ln y

ln y – ln 3 4 log y – log z

ln x – ln y 4 log (xy) – 3 log (yz)

1/3 log x 3 ln (x3y) + 2 ln (yz2)

Pre-Calculus

1/3/2007

Change of Base Formula for Logarithms

Change of Base Formula for Logarithms

bln

xlnor

blog

xlog

blog

xlogxlog

a

ab

Pre-Calculus

1/3/2007

“the” exponential function

the “natural base”

2.718281828459 (irrational, like )

Leonhard Euler (1707 – 1783)

f(x) = a • e kx for an appropriately chosen real number, k, so ek = b

exponential growth function

exponential decay function

Pre-Calculus

1/3/2007

State “exponential growth” or “exponential decay”(no calculator needed)

State “exponential growth” or “exponential decay”(no calculator needed)

a.) y = e2x b.) y = e–2x

c.) y = 2–x d.) y = 0.6–x

k > 0, exponential growth k < 0, exponential decay

b>1 so growth, but reflect over y-axis, so decay

0>b>1 so decay, but reflect over y-axis, so growth

Pre-Calculus

1/3/2007

Rewrite with e; approximate k to the nearest tenth. Rewrite with e; approximate k to the nearest tenth.

a.) y = 2x b.) y = 0.3x

y = e0.7x y = e–1.2x

e? = 2 e? = 0.3

Pre-Calculus

1/3/2007

Characteristics of a Basic Logistic Function: Characteristics of a Basic Logistic Function:

Domain:

Range:

Continuity:

Symmetry:

Boundedness:

Extrema:

Asymptotes:

End Behavior:

( - , )

( 0, 1 )

continuous

about ½, but not odd or even

B = 0, b = 0

none

y = 0, 1

lim f(x) x = 1

lim f(x) x - = 0

Pre-Calculus

1/3/2007

Based on exponential growth models, will Mexico’s population surpass that of the U.S. and if so, when?

Based on exponential growth models, will Mexico’s population surpass that of the U.S. and if so, when?

Based on logistic growth models, will Mexico’s population surpass that of the U.S. and if so, when?

Based on logistic growth models, will Mexico’s population surpass that of the U.S. and if so, when?

What are the maximum sustainable populations for the two countries?

What are the maximum sustainable populations for the two countries?

Which model – exponential or logistic – is more valid in this case? Justify your choice.

Which model – exponential or logistic – is more valid in this case? Justify your choice.

Pre-Calculus

1/3/2007

Logarithmic Functions Logarithmic Functions inverse of the exponential function

logbn = p

bp = n

logbn = p iff bp = n

find the power

2? = 32= 5

3? = 1= 0

4? = 2= ½

5? = 5= 1

2? = 2= ½

Pre-Calculus

1/3/2007

Basic Properties of Logarithms(where n > 0, b > 0 but ≠ 1, and p is any real number)

Basic Properties of Logarithms(where n > 0, b > 0 but ≠ 1, and p is any real number)

logb1 = 0 because b0 = 1

logbb = 1 because b1 = b

logbbp = p because bp = bp

blogbn = n because logbn = logbn

Example

log51 = 0

log22 = 1

log443 = 3

6log611 = 11

Pre-Calculus

1/3/2007

Evaluating Common Log ExpressionsEvaluating Common Log Expressions

log 100 =

10 log 8 =

Without a Calculator:

log 32.6 =

log 0.59 =

log (–4) =

With a Calculator:

log 710 =

2

1/7

8

1.5132176

–0.22914…

undefined

Pre-Calculus

1/3/2007

Solving Simple Equations with Common Logs and ExponentsSolving Simple Equations with Common Logs and Exponents

Solve:

10 x = 3.7

x = log 3.7

log x = – 1.6

x ≈ 0.57

x = 10 –1.6

x ≈ 0.03

Pre-Calculus

1/3/2007

Evaluating Natural Log ExpressionsEvaluating Natural Log Expressions

log e7 =

e ln 5 =

Without a Calculator:

ln 31.3

ln 0.39

ln (–3)

With a Calculator:

ln 3e = 1/3

7

5

≈ 3.443

≈ – 0.9416

= undefined

Pre-Calculus

1/3/2007

Solving Simple Equations with Natural Logs and Exponents

Solving Simple Equations with Natural Logs and Exponents

Solve:

ln x = 3.45

x = e 3.45

ex = 6.18

x ≈ 31.50

x = ln 6.18

x ≈ 1.82

Pre-Calculus

1/3/2007

Logarithmic Functions Logarithmic Functions

≈ 0.91 ln x

•vertical shrink by 0.91

xln3ln

1

3ln

xln xln

)3/1ln(

1

)3/1ln(

xln ≈ – 0.91 ln x

•reflect over the x-axis•vertical shrink by 0.91

xlogxlog bb/1

Pre-Calculus

1/3/2007

f(x) = log4x

f(x) = log5x

f(x) = log7(x – 2)

Graph the function and state its domain and range:

f(x) = log3(2 – x)

0.721 ln x

0.621 ln x

0.514 ln (x – 2)

0.091 ln (–(x – 2)

Vertical shrink by 0.721

Vertical shrink by 0.621

Vertical shrink by 0.514, shift right 2

Vertical shrink by 0.091Reflect across y-axisShift right 2

Pre-Calculus

1/3/2007

Logarithmic Functions Logarithmic Functions

one-to-one functions

u = v

isolate the exponential expression

81.22ln

7ln

take the logarithm of both sides and solve

2x = 25

x = 5

log22x = log27

x = log27

Pre-Calculus

1/3/2007

Newton’s Law of CoolingNewton’s Law of CoolingAn object that has been heated will cool to the temperature of the medium in which it is placed (such as the surrounding air or water). The temperature, T, of the object at time, t, can be modeled by:

where Tm = temp. of surrounding medium T0 = initial temp. of the object

Example: A hard-boiled egg at temp. 96 C is placed in 16 C water to cool. Four (4) minutes later the temp. of the egg is 45 C. Use Newton’s Law of Cooling to determine when the egg will be 20 C.

ktm0m eTTT)t(T

Pre-Calculus

1/3/2007

Compound Interest Compound Interest

Interest Compounded Annually Interest Compounded Annually

A = P (1 + r)t

A = Amount P = Principal r = Rate t = Time

Interest Compounded k Times Per Year Interest Compounded k Times Per Year

A = P (1 + r/k)kt

k = Compoundings Per Year

Interest Compounded ContinuouslyInterest Compounded Continuously

A = Pert

Pre-Calculus

1/3/2007

Annual Percentage YieldAnnual Percentage Yield

Annual Percentage YieldAnnual Percentage Yield

APY = (1 + r/k)k – 1

Compounded ContinuouslyCompounded Continuously

APY = er – 1

Pre-Calculus

1/3/2007

Annuities Annuities

R = Value of Paymentsi = r/k = interest rate per compoundingn = kt = number of payments

Future Value of an AnnuityFuture Value of an Annuity

n(1 i) 1FV R

i

# 11(p. 324) $14,755.51$14,755.51

Pre-Calculus

1/3/2007

Annuities Annuities

R = Value of Paymentsi = r/k = interest rate per compoundingn = kt = number of payments

Present Value of an AnnuityPresent Value of an Annuity

n1 (1 i)PV R

i

For loans, the bank uses a similar formula

Pre-Calculus

1/3/2007

Annuities Annuities

n1 (1 i)

PV Ri

12(5)0.039

1 112

27500 R0.03912

If you loan money to buy a truck for $27,500, what are the monthly pay-ments if the annual percentage rate (APR) on the loan is 3.9% for 5 years?

1 0.8231

27500 R0.00325

R $505.29