Exponential and Logarithmic Functions

Exponential Functions Example: Graph the following equations… a) b)

An exponential function is a function of the form where a is a positive real number (a > 0) and a 1. The domain of f is the set of all real numbers.

(0, 1) (1, 3)

(0, 1) (-1, 3)

(0, 3) (-1, 5) y = 2

Horizontal Asymptote: y = 2 Range: { y | y >2 } or (2, Domain: All real numbers

Properties of the Logarithm Function

Exponential Growth or Decay The amount N after t time periods (usually years) due to an initial amount, where r is the rate of growth (or decay) can model exponential growth (or decay) with the following formula:

Interest is the money paid for the use of money. The total amount borrowed is called the principal. The rate of interest, expressed as a percent, is the amount charged for the use of the principal for a given period of time, usually on a yearly (per annum) basis. Simple Interest Formula I = Prt

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

(a) Annually (b) Monthly Suppose your bank pays 4% interest per year. If $500 is deposited, how much will you have after 3 years if interest is compounded …

Formula: Continuous Compounding Interest The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is

Suppose your bank pays 4% interest per year. If $500 is deposited, how much will you have after 3 years if interest is compounded continuously?

Compound Interest Problem A Goofy Situation Goofy made purchases on his Magic Kingdom Mastercard totaling $1200. The interest is compounded monthly on his card, at a rate of 19.5%. If Goofy forgets to pay of his credit card balance (as he usually does), how much will he owe after 2 years?

Compound Interest Problem Mickey Mouse puts $400 into the bank. He chose the EPCOT Savings and Loan, which is owned by Scrooge McDuck, where interest is compounded continuously. What interest rate was offered by Scrooge, if Mickey had $432 in his account after 6 years?

Compound Interest Problem Donald Duck invests his money in MGM Studios Credit Union. The Credit Union compounds interest daily, at a rate of 2%. How long will it take Donald to double his money?

NNe kt  0 Formula: Exponential Growth and Decay (involving ‘e’)

Present and Future Value of an Annuity An Annuity is a series of equal payments made at equal intervals of time. The present value of an annuity,, is the sum of the present values of all the periodic payments (P). The lump-sum investment of dollars now will provide payments of P dollars for n periods. This formula can be used to find the present value, where i is for interest, n is for the number of payments

Present and Future Value of an Annuity The future value of an annuity,, is the sum of all of the annuity payments plus any accumulated interest. This formula can be used to find the future value, where i is for interest, n is for the number of payments Note: APR stands for “Annual Percentage Rate”

Present and Future Value of an Annuity Example: What is the present value of an annuity that pays out $2500 every week for 3 years if the APR is 6%?

Present and Future Value of an Annuity Example: What are the monthly payments on a $11,000 car over 6 years with an APR of 10%?

Present and Future Value of an Annuity Example: In order to have $50,000 at the end of 10 years, what amount should you deposit each month into an account with an APR of 7.5%?

Present and Future Value of an Annuity Example: How much money would the state of Pennsylvania need to have in an account with an APR of 11% in order to pay out $500,000 to a state Lotto winner in equal annual payments over the next 15 years?

(0, 1) (1, 0) 0 < a < 1

(1, 0) (0, 1) a > 1

1. The x-intercept of the graph is 1. There is no y-intercept. 2. The y-axis is a vertical asymptote of the graph. 3. A logarithmic function is decreasing if The graph is smooth and continuous, with no corners or gaps.

Simplify the following expressions: Solve the following equations:

Properties of Logarithms

No Solution

Logarithms and Antilogs Logarithms with base 10 are called Common Logarithms. Antilogarithm: if log x = a, then x = antilog a. Or, you can say that x = 10^a.

Logarithms and Antilogs Logarithms with base ‘e’ are called Natural Logarithms and are written as ‘ln’ Anti-natural logarithm: if ln x = a, then x = antiln a. Or, you can say that x = e^a.

Change in Base Formula If a, b, and n are positive numbers and neither a nor b is 1, then the following formula holds true:

How long will it take to double an investment earning 10% per year compounded quarterly?

How long will it take to double an investment earning 10% per year compounded continuously?

Uninhibited Growth and Decay Formula Where n is the initial amount, y is the final amount, t is for time, and k is a constant (depending on the substance that is growing or decaying).

NtNe kt ()  0 k > 0 Uninhibited Growth (cells, bacteria, etc.)

NNe kt  0 Exponential Growth or Decay Formula (in terms of e) for growth, k > 0 for decay, k < 0

Uninhibited Growth Example: For a certain strain of bacteria, k =.0325 and t is measured in days. How long will it take 25 bacteria to increase to 500?

AtAe kt ()  0 k < 0 Uninhibited Decay

Uninhibited Radioactive Decay Example: The half-life of a certain radioactive substance is 500 years. If 300 kg of the substance is present now, how much will be present in 800 years?

The half-life of Uranium-234 is 200,000 years. If 50 grams of Uranium-234 are present now, how much will be present in 1000 years. NOTE: The half-life is the time required for half of the radioactive substance to decay.

Uninhibited Radioactive Decay Example: If 18 grams of a certain radioactive substance decays to 5 grams after 340 years, what is its half-life?

The half-life of a certain radioactive substance is 2500 years. If 35 grams of the substance are present now, how much will be present in 1500 years?

Newton’s Law of Cooling  utTuTek kt ()  0 0 T : Temperature of surrounding medium u o : Initial temperature of object k : A negative constant

A cup of hot chocolate is 100 degrees Celsius. It is allowed to cool in a room whose air temperature is 22 degrees Celsius. If the temperature of the hot chocolate is 85 degrees Celsius after 4 minutes, when will its temperature be 60 degrees Celsius?

Newton’s Law of Cooling y : temperature of cooling object c: Temperature of surrounding medium a: initial temp – temp of surrounding medium k : a constant t : time it takes for object to cool

Newton’s Law of Cooling Example: Find the time it takes a cup of soup to cool to 100 degrees, if it has been heated to 180 degrees. Room temperature is 70 degrees and k =.01.

Newton’s Law of Cooling: Time of Death At 4:45 AM, the CSI Team was called to the home of a person who had died during the night. In order to estimate the time of death, Grissom took the person’s body temperature twice. At 5:00 AM, the body temp was 85.7  F and at 5:30, the temp was 82.8  F. The room temperature was 70 . Find the approximate time of death.

Logarithmic Application: pH level of a chemical solution based on the hydrogen ions present pH = -log 10 [H+] Example: The pH of a solution is 5.3. Find the hydrogen ion concentration.

Logarithmic Application: pH level of a chemical solution based on the hydrogen ions present pH = -log 10 [H+] Example: Find the pH level for a certain solution with a hydrogen ion concentration of.00024

Logarithmic Application: Decibel Level of Sound (D) based on the intensity (I) of its sound waves ( in watts per square meter) Example: Find the number of Decibels if the intensity of the sound generates watts per square meter at a distance of 12 feet.

Logarithmic Application: Decibel Level of Sound (D) based on the intensity (I) of its sound waves ( in watts per square meter) Example: At a recent concert, the Decibel level registered at 95 decibels. Find the intensity in watts per square meter of the sound.

Logarithmic Application: Earthquake Magnitude (M) on the Richter Scale based on seismographic reading in millimeters (x) divided by a zero-level earthquake ( ) Example: Find the Richter Scale reading for an earthquake 75,342 millimeters, 100 kilometers from the center of the quake.

Logarithmic Application: Earthquake Magnitude (M) on the Richter Scale based on seismographic reading in millimeters (x) divided by a zero-level earthquake ( ) Example: A recent earthquake in California measured 4.1 on the Richter Scale. What would be the magnitude of a quake 100 times stronger?

Euler’s Formula Leonhard Euler developed the following relationship between trigonometric series and exponential series. Below are the trig series for sine and cosine.

Euler’s Formula Here is the exponential series for e^x. Suppose we replace x with

Euler’s Formula

This can be re-written using the trig series shown earlier to give us Euler’s Formula: This formula can be used to write the Exponential Form of a Complex Number, X + Yi…

Euler’s Formula Now we know that there is no real number that is logarithm of Negative value. However, we can use Euler’s Formula to find A complex number that is the natural log of –1.

Euler’s Formula Example: Evaluate ln(-12).

Euler’s Formula Example: Write each complex number in Exponential Form.

Theorem Present Value Formulas The present value P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year, is If the interest is compounded continuously, then

How much should you deposit today in order to have $20,000 in three years if you can earn 6% compounded monthly from a bank C.D.?

Graph What happens to the value of u(t) as t increases without bound?

Logistic Growth Model where a, b, and c are constants with c > 0 and b > 0.

What is the carrying capacity?500 Graph the function using a graphing utility.

What was the initial amount of bacteria?

When will the amount of bacteria be 300 grams?

(1, 0) (e, 1)

(4, 0) (e + 3, 1) x = 3

Between 5:00 p.m. and 6:00 p.m. cars arrive at Wendy’s drive-thru at the rate of 15 cars per hour (0.25 cars per minute). The following formula from statistics can be used to determine the probability a car will arrive within t minutes of 5:00 p.m. Determine the probability the car will arrive within 5 minutes of 5:00 p.m. (that is, before 5:05 p.m.)

Determine the probability the car will arrive within 30 minutes of 5:30 p.m. (that is, before 5:30 p.m.) Graph F(t) using your graphing utility. What value does F approach as t becomes unbounded in the positive direction?

Growth and Decay in Nature Final count is y, initial count is, c is the constant of proportionality, t is time, and T is time per cycle of c

Growth and Decay in Nature Example: A certain population of single-celled organisms doubles every 3 days. If the initial amount of organisms was 1200 and now there are 100,000, how many days have passed?

Properties of Logarithms

Exponential Functions: Application Problems 1. For a certain radioactive material, the half life is 1200 years. How much of a 400 gram substance would remain after 500 years? 2. What are the monthly payments on a mortgage for a $250,000 house over 20 years with APR of 8%? 3. How long would it take for an investment of $300 to reach $1500, if the interest was compounded daily at 6.5%?

Rational Exponents Example: Express using Rational Exponents Example: Express using Radicals

Rational Exponents Example: Simplify the following… a) b) c)