3.1 Exponential FunctionsExponential function – any function whose equation contains a variable in the exponent. [measures rapid increase or decrease (Example: epidemic growth)]

f(x) = bx

f – exponential function b - constant base (b > 0, b 1) x = any real number -- domain is (-∞ , ∞)

f(x) = 2x g(x) = 10x h(x) = 3x+1

32+1 = 33 = 2732 = 92

31+1 = 32 = 931 = 31

30+1 = 31 = 330 = 10

3-1+1 = 30 = 13-1 = 1/3-1

3-2+1 = 3-1 = 1/33-2 = 1/9-2

g(x) = 3x+1f (x) = 3xx

f (x) = 3xg(x) = 3x+1

(0, 1)(-1, 1)

1 2 3 4 5 6-5 -4 -3 -2 -1

Graphing Exponential Functions:

Shift up c unitsf(x) = bx + c

Shift down c unitsf(x) = bx – c

Shift left c unitsf(x) = bx+c

Shift right c unitsf(x) = bx-c

All properties above are the sameExcept #4 which is ‘decreasing’

See P.193

3.2 Investment and Interest ExamplesCompound InterestKaren Estes just received an inheritance of $10000And plans to place all money in a savings accountThat pays 5% compounded quarterly to help her sonGo to college in 3 years. How much money will beIn the account in 3 years?

Use the formula:A = P(1 + r/n)nt

A = amount in account after t yearsP = principal or amount investedt = time in yearsr = annual rate of interestn = number of times compounded per year•Annual => n = 1•Semiannual => n = 2•Quarterly => n = 4•Monthly => n = 12•Daily => n = 365

Simple InterestJuanita deposited $8000 in a bank for 5 years at a simple interest rate of 6%

1. How much interest will she get?2. How much money will she have at the end of 5 years

Use the formula:I = Prt

I = (8000)(.06)(5) = $2400 (interest earned in 5 years)

In 5 years she will have a total ofA = P + I = 8000 + 2400 = $10,400

As n increases (n ∞), A does NOT increase indefinitely.It gets closer and closer to a fixed number “e”, and gives us the continuous compound interest formula: A = Pert

Compound Interest related to Simple InterestIf a principal of P dollars is borrowed for a period of t years at a perannum interest rate r, the interest I charged is:

I = PrtAnnually -> Once per year Semiannually -> Twice per yearQuarterly -> Four times per year Monthly -> 12 times per yearDaily -> 365 times per year

Example: A credit union pays interest of 8% per annum compounded quarterly on aCertain savings plan. If $1000 is deposited in such a plan and the interest is Left to accumulate, how much is in the account after 1 year?

I = (1000)(.08)(1/4) = $20 => New principal now is: $1020I = (1020)(.08)(1/4) = $20.40 => New principal now is: $1040.40I = (1040.40)(.08)(1/4) = $20.81 => New principal now is: $1061.21I = (1061.21)(.08)(1/4) = $21.22 => New principal now is: $1082.43

Compound Interest Formula: The amount, A, after t years due to a principal PInvested at an annual interest rate r compounded n times per year is:

A = P (1 + r/n)nt

The Natural Base eAn irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately,

The number e is called the natural base. The function f (x) = ex is called the natural exponential function.

-1

f (x) = ex

f (x) = 2x

f (x) = 3x

(0, 1)

(1, 2)

1

2

3

4

(1, e)

(1, 3)

Analogous to the continuous compound interest formula, A = Pert

The model for continuous growth and decay is: A(t) = A0ekt

A(t) = the amount at time tA0 = A(0), the initial amountk = rate of growth (k>0) or decay (k < 0)

In year 2000, population of the world was about 6 billion; annual rate of growth 2.1%. Estimate the population in 2030 and 1990.

2000=year 0, 2030=year 30, year 1990=year -10

A(30) = 6e(.021)(30) = 11.265663 (billion people)A(-10) = 6e(.021)(-10) = 4.8635055 (billion people)

** Note these are just estimates. The actual population in 1990 was 5.28 billion

Exponential Growth & Decay(Law of Uninhibited Growth/Decay)

A0 is the original amount of ‘substance’ (drug, cells, bacteria, people, etc…)

3 -1Solve: 2 32x 42 13

1Solve: x x

xe e

e

Solving Exponential Equations

25 = 32, so

23x-1 = 25

3x – 1 = 5

3x = 6

X = 2

e2x-1 = e-4x

e3x

e2x-1 = e-7x

2x – 1 = -7x

-1 = -9x

X = 1/9

3.3 Logarithmic FunctionsA logarithm is an exponent such that for b > 0, b 1 and x > 0 y = logb x if and only if by = x

Logarithmic equations Corresponding exponential forms1) 2 = log5 x 1) 52 = x2) 3 = logb 64 2) b3 = 643) log3 7 = y 3) 3y = 74) y = loge 9 4) ey = 9

log25 5 = 1/2 because 251/2 = 5.25 to what power is 5?log25 5

log3 9 = 2 because 32 = 9.3 to what power is 9?log3 9

log2 16 = 4 because 24 = 16.2 to what power is 16?log2 16

Logarithmic Expression Evaluated

Question Needed for Evaluation

Evaluate the

Logarithmic Expression

Logarithmic Properties

Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b).

Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).

logb bx = x The logarithm with base b of b raised to a power equals that power.b logb x = x b raised to the logarithm with base b of a number equals that number.

Graphs of f (x) = 2x and g(x) = log2 x [Logarithm is the inverse of the exponential

function]

4

2

8211/21/4f (x) = 2x

310-1-2x

2

4

310-1-2g(x) = log2 x

8211/21/4x

Reverse coordinates.

-2 -1

6

2 3 4 5

5

4

3

2

-1-2

6

f (x) = 2x

f (x) = log2 x

y = x

Properties of f(x) = logb x

•Domain = (0, ∞)•Range = (-∞, +∞)•X intercept = 1 ; No y-intercept•Vertical asymptote on y-axis•Decreasing on 0<b<1; increasing if b>1•Contains points: (1, 0), (b, 1), (1/b, -1)•Graph is smooth and continuous

Common Logs and Natural Logs

A logarithm with a base of 10 is a ‘common log’log10 1000 = ______ because 103 = 1000

If a log is written with no base it is assumed to be 10.

log 1000 = log10 1000 = 3

3

A logarithm with a base of e is a ‘natural log’loge 1 = ______ because e0 = 1

If a log is written as ‘ln’ instead of ‘log’ it is a natural log

ln 1 = loge 1 = 0

0

Doubling Your InvestmentsHow long will it take to double yourMoney if it earns 6.5% compoundedcontinuously?

Recall the continuous compound interest formula: A = Pert

If P is the principal and we want P to double, the amount A will be 2P.

2P = Pe(.065)t

2 = e(.065)t

Ln 2 = ln e(.065)t

Ln 2 = .065t

T = ln 2 = .6931 ≈ 10.66 ≈ 11 years .065 .065

At what rate of return compounded continuously would your money double in 5 years?

A = Pert

2P = Per(5)

2 = er(5)

ln 2 = ln e5r

.6931 = 5r

r = .6931/5R = .1386

So, annual interest rate needed is: 13.86%

Newton’s Law of CoolingT = Ts + (T0 – Ts)e-kt T = Temperature of object a time t

Ts = surrounding temperatureT0 = T at t = 0

A McDonald’s franchise discovered that when coffee is poured from a CoffeeMakerwhose contents are 180° F into a noninsulated pot, after 1 minute, the Coffee cools to 165° F if the room temperature is 72° F. How long should employees wait before pouring coffee from this noninsulated pot into cups to deliver it to customers at 125°F?

T0 = 180 and Ts = 72 Now, T = 72 + 108e-.1495t

T = 72 + (180 – 72) e-kt 125 = 72 + 108e-.1495t

T = 72 + 108e-kt 53 = 108e-.1495t

.4907 = e-.1495t

Since T = 165 when t = 1 ln .4907 = ln e-.1495t

165 = 72 + 108e-k(1) -.7119 = -.1495t 93 = 108e-k t = 4.7693 = 108e-k Employees should wait about 5 minLn .8611 = ln e-k to deliver the coffee at the desired-.1495 = -k temperature.K = .1495

3.4 Properties & Rules of LogarithmsBasic PropertiesLogb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b).

Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).Inverse Properties

logb bx = x The logarithm with base b of b raised to a power equals that power.b logb x = x b raised to the logarithm with base b of a number equals that number.

logb(MN) = logb M + logb N Product Rule

Quotient Rule

logb M = logb M - logb N N

logb M = p logb M Power Rule

p

For M>0 and N > 0

Logarithmic Property Practice

logb(MN) = logb M + logb N

1) log3 (27 • 81) =

2) log (100x) =

3) Ln (7x) =

logb M = logb M - logb N N

logb M = p logb M p

1) log8 23 x

2) Ln e5

11

1) log5 74

2) Log (4x)5

3) Ln x2 =4) Ln x =

=

=

=

=

Product Rule

Power Rule

Quotient Rule

Expanding Logarithmslogb(MN) = logb M + logb N

logb M = logb M - logb N N

logb M = p logb M p

2) log6 3 x36y4

1) Logb (x2 y ) 3) log5 x25y3

4) log2 5x2

3

Condensing Logarithmslogb(MN) = logb M + logb N

logb M = logb M - logb N N

logb M = p logb M p

1) log4 2 + log4 32 4) 2 ln x + ln (x + 1)

2) Log 25 + log 4 5) 2 log (x – 3) – log x

3) Log (7x + 6) – log x 6) ¼ logb x – 2 logb 5 – 10 logb y

Note: Logarithm coefficientsMust be 1 to condense. (Use power rule 1st)

The Change-of-Base Property

Example: Evaluate log3 7

Most calculators only use:

• Common Log [LOG] (base 10)• Natural Log [LN] (base e)

It is necessary to use the changeOf base property to convert toA base the calculator can use.

Matching Data to an Exponential Curve

Find the exponential function of the form f(x) = aebx that passes through the points (0,2) and (3,8)

F(0) = 2 f(3) = 82 = aeb(0) 8 = aeb(3) f(x) = 2e((ln4)/3)x

2 = ae0 8 = 2e3b

2 = a(1) 4 = e3b

a = 2 ln 4 = ln e3b

ln 4 = 3bb = (ln4)/3

3.5 Exponential & Logarithmic Equations

Step 1: Isolate the exponential expression.Step 2: Take the natural logarithm on both sides of the equation.Step 3: Simplify using one of the following properties:

ln bx = x ln b or ln ex = x.

Step 4: Solve for the variable using proper algebraic rules.

54x – 7 – 3 = 10log4(x + 3) = 2. 3x+2-7 = 27

Examples (Solve for x):

log 2 (3x-1) = 18

More Equations to Try

3 3Solve: log 4 2log x

2 2Solve: log 2 log 1 1x x

Solve: 3 7x

Solve: 5 2 3x

1 2 3Solve: 2 5x x

Solve: 9 3 6 0x x