Exponential FunctionsExponential Functions have the form: Example: Complete the tables for and graph:a) for x>0

b) for x>0

x 0 1 2 3 4y

x 0 1 2 3 4y

Exponential Growth and Decay• Exponential Growth: As x increases, y increases.• Exponential Decay: As x increases, y decreases.• For the function:

If a>0 and b>1 we have exponential growthIf a >0 and 0<b<1 we have exponential decay

Asymptote: A line (axis) that a function approaches as x or y approaches its extreme values. It can be vertical or horizontal.What are the asymptotes of ? What is the y-intercept of ?

Applications• Exponential Growth:– Compound Interest– Population Growth

• Exponential Decay:– Time Dependent Heat Transfer– Attenuation of Light traveling through a medium– Time dependence of Voltage/Current in electronic

circuits– Radioactive processes/ Half Life– Probability dependence of small particles (Quantum

Physics)

Modeling Exponential Growth and Decay

• Quantities that change exponentially often do so as functions of time. So we talk about “growth rates” and “decay rates”.

• We can rewrite the function as where:if r>1, then b = growth factor and r = growth rateif r<1, then b = decay factor and r = decay rate.

As a function of time: where: A(t) = amount at some time (t)a = initial amount (when t=0)r = rate of growth or decayt = time

Example 1: Population Growth (Humans)

• The population of Jacksonville was 3,810 in 2007 and is growing at an annual rate of 3.5%. At this growth rate, what should its population be in 2020?What is the growth rate, expressed as a decimal?What is the growth factor?What is the initial population?What is the value of time (t) we need to use?What is the population at this time t?

Example 2: Compound Interest

• You invested $1000 in a bank account that pays 5% annual interest. How much will be in the account after 6 years?What is the growth rate, expressed as a decimal?What is the growth factor?What is the initial amount?What is the value of time (t) we need to use?What is the amount at this time t?

• How many years will it take for the account to have at least $1500 in it? (Use a graphing calculator and the intersect function)

How do we Transform Exponential Functions?

• Recall: For a Parent Function: f(x)• We transform as: g(x) = af(x-h) + k• Applied to the exponential function: f(x) = bx

• We have: g(x) = ab(x-h) + k• Using your graphing calculator: Graph:

f1(x) = 2x

f2(x) = 2(x-4)

f3(x) = 3·2(x+4) + 3

How can we Model Data using Exponential Functions?

Example: Cooling Coffee• Coffee is often brewed at around 200 ◦F and is considered

cool enough to drink at 185◦F.

• Considering the given data, how much time should we wait before drinking coffee?

• Assumption: Coffee cools down eventually to 68◦F.

Time (min)

0 5 10 15 20 25 30

Temp (◦F)

203 177 153 137 121 111 104

Coffee Cooling cont’d1) Enter data into lists: time, temp 2) Plot the data to see if an exponential model makes sense. (If

you do exponential linear regression on this plot it will be incorrect because calculator assumes a zero asymptote; Try it.)

3) Make a third list: tempshift = temp-68 (to adjust asymptote), Plot tempshift vs. time.

4) Do an exponential linear regression on this graph: f(x) = 134.5 (.956)x

5) Shift up by 68 to adjust asymptote and obtain our cooling function:

T(t) = 134.5(.956)t + 686) Plot the cooling function and y = 185 to find the time it

takes for the coffee to be cool enough to drink.

Compound Interest Revisited• An account starts with $1.00 and pays 100 percent

interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

• If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding:$1.00×1.52 = $1.00× = $2.25 at the end of the year.

• Compounding quarterly yields $1.00×1.254 = $1.00× = $2.4414 at year’s end.

Compound Interest cont’d

• So if = the number of times per year the interest is compounded, and we get: from a $1 initial investment.

Percentage wise, we get:

We calculate it like this: We get:

1 100% applied once

2 50% applied twice

4 25% applied four times

10 10% applied ten times

100 1% applied 100 times

1000 0.1% applied 1000 times

Compound Interest cont’d

• So if = the number of times per year the interest is compounded, and we get: from a $1 initial investment.

Percentage wise, we get:

We calculate it like this: We get:

1 100% applied once $2.000

2 50% applied twice $2.250

4 25% applied four times

$2.441

10 10% applied ten times

$2.594

100 1% applied 100 times

$2.705

1000 0.1% applied 1000 times

$2.717

Euler’s Number () (Discovered by Bernoulli)

• We see that as gets higher and higher, the value of approaches a constant value:

• This number is irrational.• In , “” is referred to as the “base”.• From this definition of we find many functions

of the form: . Such functions are said to have a “natural base”. ( comes up in nature a lot or maybe because the slope of at any point is (That is the slope of this function is itself!).

Continuously Compounded Interest

• Using the definition of and some calculus, we can arrive at a formula for finding how much money you’ll have in an account at any point in time A(t), given:

P = principal = initial investment.r = interest ratet = time in years

Continuously Compounded Interest

• Suppose you put $3000 into an account paying 5% interest compounded continuously.– How much would you have after 4 years?– How much would you have after 8 years?

• How do the numbers above compare to how much you would have if the interest were compounded annually?

• How can we use a graphing calculator to compare the growth of a principal for different interest rates? Try comparing 5% to 8%. Does a small change interest rate matter much over the years?

“Hwk 33”

• Pages 447-450• 21, 22, 31, 32, 41, 36

Logarithmic Functions: The Inverse of the Exponential Function

• The exponential function: has the inverse: which satisfies the following definition:For :

We say “log base b of x”.

Logarithmic and Exponential Forms of an Equation

• We can write a logarithmic equation in exponential form:

If then

• We can write an exponential equation in logarithmic form:

If then • Examples:

1. What is the logarithmic form of 100 = 102 ?

2. Write the following as an exponential equation and find :

More Examples So: If then If then

• Find the logarithmic form ofa) 81 = 34

b)

c) 1=30

• Evaluate each logarithm:a) log5125

b) log4 32

c) log64

Logarithm Bases• a commonly used base is base 10, called the

“common logarithm”. Since it is so often used, it is assumed unless a different base is specified.

• Another commonly used base is , called the “natural logarithm” , designated with the symbol .

• Examples of notation: = logarithm of in base 10 = logarithm of in base 3 = logarithm of in base called “the natural log of ”.

Logarithmic ScalesWhen a function describing a quantity varies over a very large range, we often describe it by referring to its logarithm.Example: Earthquake Magnitude: the Richter Scale• Uses the logarithms (base 10) of wave amplitudes to

compare strengths of earthquakes.• An increment by 1 on the scale corresponds to a tenfold

increase in “intensity” (amplitude size). (This actually corresponds to an increase in released energy of about 31 times).

• An Earthquake measuring 4.0 on the Richter scale is:• Ten times more intense than one measuring 3.0 and• One hundred times more intense than one measuring 2.0.

Richter Scale ExampleThe disastrous December 2004 Sumatra earthquake measured 9.3 on the Richter scale. The following March, another quake hit, which measured 8.7. How much more intense was the first earthquake?Let be the intensity of the 12/04 quake.Let be the intensity of the 3/05 quake.Then and So: and and times more intense.

Another Richter Scale Example• In 1995, Mexico received an 8.0 earthquake

and in 2001, Washington state received one of magnitude 6.8. How many times more intense was the quake that hit Mexico?

The pH Scale• The of a substance is defined as:

where = the concentration number of moles of Hydrogen ions per liter. (1 mole of ions equals 6.02×1023 ions).

• is a measure of an object’s acidity; a above 7.0 is considered alkaline (or basic) and a below 7.0 is considered acidic. If =7.0, the solution is neutral.

• What is the concentration of a neutral solution?• For each of the following substances, find the :

Food Apple Juice

Buttermilk Cream Ketchup Shrimp Sauce

Strained Peas

3.2×10-4 2.5×10-5 2.5×10-7 1.3×10-4 7.9×10-8 1.0×10-6

The pH Scale (cont’d)

• A sample of Seawater has a of 8.5. What is the concentration of Hydrogen ions in it?

HWK 34

• Pages 456-458: 16, 18, 33, 34, 47, 51, and the following additional problems:

1) A sample of pond water is found to have a pH of 6.4. What is the concentration of Hydrogen ions in it?

2) An cleaner has a concentration of hydrogen ions of . What is the pH of the cleaner?

Graphing Logarithmic and Exponential Functions

• is the inverse of • Example: For the function: :

1) State the domain, range, y-intercept, asymptotes.2) State the inverse function3) Graph both functions simultaneously using a graphing

calculator4) How is the function related to its inverse graphically?

(That is, what transformation would turn the function into its inverse?)

How can we Graph a Logarithmic Function using its Inverse?

• Example: 1) Write the inverse of the function2) Make a table (x,y) of easy to calculate values

for the inverse.3) Interchange the (x,y) ordered pairs; these give

points on the original function.4) Plot the interchanged ordered pairs.

How do we Transform Logarithmic Functions?

• Recall: For a Parent Function: f(x)• We transform as: g(x) = af(x-h) + k• So what series of transformations is represented

by ?• What effect does the value of have on the

transformation?• Plot simultaneously:

Properties of Logarithms

• For any positive numbers and , where , the following properties apply:

Let’s prove these properties using the properties we’ve already derived for exponentials.Starting point: Let and Then: and

Examples• Write the following as a single logarithm:

More Examples• Expand each of the following logarithms into sums

or differences of two or more logarithms:

Change of Base Formula• For positive numbers with :

• We can prove this. Start by multiplying each side by then use the properties of exponents.

Examples• Evaluate the following by hand or by using a

calculator using only base 10 logarithms:

Solving Exponential EquationsMethods Include• Express each side as an expression raised to a

common base.• Take the logarithm of both sides • Use the graphing/intersect method• Examples: Solve for :

Another Example• Your lumber company has 1,200,000 trees. You will

harvest 7% of the trees each year. How many years will it take to harvest half your trees?

1) Is an exponential model reasonable? What is the general form of an exponential model?

2) Let n = number of years to harvest, T(n) = number of trees remaining after n years. Write the specific exponential model equation.

3) Solve the model using logarithms.

Solving Logarithmic Equations

Methods Include• Casting each side of the equation as an exponent

of a common base. Then use the change of base formula to get to base 10.

• Use the graphing/intersect method• Examples: Solve for :

Natural Logarithms• The function has an inverse: where . The natural logarithm is used so much, it has its own special notation.

• The natural logarithm is simply the logarithm using Euler’s number () as the base.

• So let’s do some problems with it.

Some Examples1) Write as a single natural log:

a)

2) Solve:

a)

b)

Example: Stable Orbit about Earth(pg480)

By the way: Are they right about v=7.7 km/s giving a stable orbit at a height of 300 km above the Earth? (Earth Radius = 6.4×106m and Earth Mass = 6.0×1024kg).

Also Given: c = 2.8 km/s

BAD SLIDE.Compound Interest Revisited• Suppose we invest $1.00 at 100% annual interest. How

much we get per year depends on how often the interest is applied; i.e. (compounded) Let = the number of times per year the interest is compounded.

• If n = 1, we get 100% of our initial investment at the end of the year: 1+ 1*100% = $2

• If n=2: then we get 50% of our initial investment after year (bringing us to $1.50) and then 50% of the $1.50 at the end of the year: (1.50) (1.50) = = $2.25

• If n=4: then we get 25% after year ($1.25) then 25% again a year later: ($1.25)×(1.25) then 25% after 3rd quarter: ($1.25)(1.25)(1.25), then finally: $(1.25)(1.25)(1.25)(1.25) = = $2.44.