Exponential & Logarithmic Functions 1-to-1 Functions; Inverse Functions Exponential Functions Logarithmic Functions Properties of Logarithms; Exponential & Logarithmic Models Logarithmic and Exponential Equations Compound Interest Applications

{(1, 1), (2, 4), (3, 9), (4, 16)} one-to-one {(-2, 4), (-1, 1), (0, 0), (1, 1)} not one-to-one One-to-one Functions

Theorem: Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

Use the graph to determine whether the function is one-to-one. Not one-to-one!

Use the graph to determine whether the function is one-to-one. Is one-to-one!

Letf denote a one-to-one function yfx  (). The inverse of f, denoted by f  1, is a function such that  ffxx   1 () for everyx in the domain off and  ffxx   1 () for everyx in the domain of f  1. Inverse Functions

Domain of fRange of f

Theorem: Graphs of f & f -1 are a reflection image pair The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

y = x (2, 0) (0, 2)

Find the inverse of fx x x(),    5 3 The function is one-to-one. 3

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. Exponential Functions

(0, 1) (1, 3) (1, 6) (-1, 1/3) (-1, 1/6)

(-1, 3) (-1, 6) (0, 1)(1, 1/3)(1, 1/6)

(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

Logarithm Functions

Properties of the Logarithm Function

(0, 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.

(1, 0) (e, 1)

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

Theorem: Properties of Logarithms

Theorem: Change-of-Base Formula

Applications: Exponential Function in Biology

After drawing a scatter diagram of the data, one finds the exponential function of best fit to the data to be: Typically, one then expresses this in base e form as follows.

Express the curve in the form

Predict the population after 9 days.

The following data represent the amount of carbon emissions in millions of metric tons: Applications: Logarithim Function in Pollution Study

Then one can predict the carbon emissions in the year 2000 as: After drawing a scatter diagram of the data, one finds the exponential function of best fit to the data to be:

Applications: Solution of Logarithmic & Exponential Equations

No Solution

This is an example of what is called a transcendental equation. These equations can only be solved graphically or numerically. Estimate solution: rearrange then plot each side of Can you see the solution is about x=-1.78?

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 Applications: Solution of Interest Problems

Theorem: 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 annum. If $500 is deposited, how much will you have after 3 years if interest is compounded …

Graph What is the value of y for x = 10? What is the value of y for x = 20? Describe the behavior of the graph.

Theorem: Continuous Compounding 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 annum. If $500 is deposited, how much will you have after 3 years if interest is compounded continuously?

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.?

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

NtNe kt ()  0 k > 0 Uninhibited Growth of Cells More Applications: Solution of Logarithmic & Exponential Equations in Models

Uninhibited Radioactive Decay

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 of is the time required for half of radioactive substance to decay.

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?

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?

The intensity of a sound wave is the amount of energy the wave transmits through a given area. The least intense sound that a human ear can detect at a frequency of 100 Hertz is about watt per square meter. The loudness L(x), measured in decibels of a sound of intensity x is defined as Lx x ()log   10 12

What is the loudness of a rock concert if its intensity is 0.2 watts per square meter? Close to the threshold of pain due to sound which is about 120 db.

The Richter scale converts seismographic readings into numbers that provide an provide an easy reference for measuring the magnitude M of an earthquake.

An earthquake whose seismographic reading measures x millimeters has magnitude M(x) given by All earthquakes are compared to a zero-level earthquake whose seismographic reading measures millimeter at a distance of 100 kilometers from the epicenter. where x 0 = is the reading of a zero-level earthquake the same distance from its epicenter.

Determine the magnitude of an earthquake whose seismographic reading is 3 millimeters at a distance 100 kilometers from the epicenter. The earthquake measures 3.5 on the Richter scale.