# Exponential and Logarithmic Functions

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Exponential and Logarithmic Functions
Chapter 13 Exponential and Logarithmic Functions

Section 13.1: Exponential Functions and Their Graphs
MAT 205 SP 2009 Section 13.1: Exponential Functions and Their Graphs Definition of an Exponential Function The exponential function with base b is denoted by So, in an exponential function, the variable is in the exponent.

Exponential Functions
Which of the following are exponential functions?

Graphs of Exponential Functions
They can be broken into two categories— exponential growth, and exponential decay (decline).

The Graph of an Exponential Growth Function
We will look at the graph of an exponential function that increases as x increases, known as the exponential growth function. It has the form Example: y = 2x x y -5 -4 -3 -2 -1 1 2 3 y = 2x Notice the rapid increase in the graph as x increases The graph increases slowly for x < 0. y-intercept is (0, 1) Horizontal asymptote is y = 0.

The Graph of an Exponential Decay (Decline) Function
We will look at the graph of an exponential function that decreases as x increases, known as the exponential decay function. It has the form Example: y = 2-x y = 2-x x y -3 -2 -1 1 2 3 4 5 Notice the rapid decline in the graph for x < 0. The graph decreases more slowly as x increases. y-intercept is (0, 1) Horizontal asymptote is y = 0.

Graphs of Exponential Functions
Notice that f(x) = 2x and g(x) = 2-x are reflections of one another about the y-axis. Both graphs have y-intercept ___________ and horizontal asymptote ________ . The domain of f(x) and g(x) is _________; the range is _______.

Graphs of Exponential Functions
MAT 205 SP 2009 Graphs of Exponential Functions Also, note that , applying the properties of exponents. So an exponential function is a decay function if The base b is greater than one and the function is written as f(x) = b-x -OR- The base b is between 0 and 1 and the function is written as f(x) = bx

Graphs of Exponential Functions
MAT 205 SP 2009 Graphs of Exponential Functions Examples: In this case, b = 0.25 (0 < b < 1). In this case, b = 5.6 (b > 1).

Natural base e It may seem hard to believe, but when working with exponents and logarithms, it is often convenient to use the irrational number e as a base. The number e is defined as This value approaches as x approaches infinity.

Evaluating the Natural Exponential Function
To evaluate the function f(x) = ex, we will use our calculators to find an approximation. You should see the ex button on your graphing calculator (Use ). Example: Given , find f(3) and f(-0.5) to 3 decimal places. ≈ ____________ ≈ _______________

Graphing the Natural Exponential Function
Growth or decay? Domain: Range: Asymptote: x-intercept: y-intercept: List four points that are on the graph of f(x) = ex.

Graphing the Natural Exponential Function
Determine the following: Growth or decay? Domain: Range: Asymptote: x-intercept: y-intercept:

Example The population of a town is modeled by the function where t = 0 corresponds to 1990 and P is the town’s population in thousands. According to the model, what was the town’s population in 1990? According to the model, what was the town’s population in 2008?

MAT 205 SP 2009 Example (continued) Graph the function on your calculator and determine in which year the town’s population reached 75,000 people. How would we solve this algebraically??

Section 13.2: Logarithmic Functions
MAT 205 SP 2009 Section 13.2: Logarithmic Functions Now that you have studied the exponential function, it is time to take a look at its INVERSE: the LOGARITHMIC FUNCTION. In the exponential function, the independent variable was the exponent. So we substituted values into the exponent and evaluated it for a given base. For example, for f(x) = 2x f(3) =

Logarithmic Functions
MAT 205 SP 2009 Logarithmic Functions For the inverse (logarithmic) function, the base is given and the answer is given, so to evaluate a logarithmic function is to find the exponent. That is why I think of the logarithmic function as the “Guess That Exponent” function. Warm Up: Give the value of ? in each of the following equations.

Logarithmic Functions (continued)
For example, to evaluate log28 means to find the exponent such that 2 raised to that power gives you 8.

Logarithmic Functions (continued)
The following definition demonstrates this connection between the exponential and the logarithmic function. Definition of an Logarithmic Function For y > 0, b > 0, and b ≠ 1, If y = bx , then x = logby y = bx is the exponential form x = logby is the logarithmic form We read logby as “log base b of y”.

MAT 205 SP 2009 Subliminal Message: The exponential and logarithmic functions of the same base are inverses.

Converting Between Exponential and Logarithmic Forms
If y = bx, then x = logby I. Write the logarithmic equation in exponential form. a) b) II. Write the exponential equation in logarithmic form.

The plan is to convert to exponential form.
Evaluating Logarithms w/o a Calculator To evaluate logarithmic expressions by hand, we can use the related exponential expression. Example: Evaluate the following logarithms: The plan is to convert to exponential form.

Evaluating Logarithms w/o a Calculator (cont.)

Evaluating Logarithms w/o a Calculator
Okay, try these.  e) f) g) h)

Determine the value of the unknowns
a) b)

Determine the value of the unknowns
c) d)

Graphs of Logarithmic Functions
Example: Graph f(x) = 2x and g(x) = log2x in the same coordinate plane. Solution: To do this, make a table of values for f(x) and then switch the x and y coordinates to make a table of values for g(x).

Graphs of Logarithmic Functions (continued)
Inverse functions f(x) = 2x g(x)= log2x y =x f(x) = 2x g(x) = log2x x f(x) -4 1/16 -2 1/4 1 2 4 16 x g(x) 1/16 -4 1/4 -2 1 4 2 16

Graphs of Logarithmic Functions (continued)
Notice how the domain and range of the inverse functions are switched. The exponential function has Domain: ____________ Range: ____________ Horizontal asymptote: _________ The logarithmic function has Domain: __________ Range: ___________ Vertical asymptote: __________ f(x) = 2x g(x)= log2x y =x

Back in my day, we used log tables and slide rules!”
Section 13.4: Evaluating Common Logarithms with a Calculator Not all logarithmic expressions can be evaluated easily by hand. In fact, most cannot. For example, to evaluate is to find x such that 2x = 175. This is not a simple task. In fact, the answer is irrational. For these types of problems, we will use the calculator. “Calculators?? Back in my day, we used log tables and slide rules!”

Evaluating Common Logarithms with a Calculator (continued)
The calculator, however, only calculates two different base logarithms—the common logarithm and the natural logarithm. I. The COMMON LOGARITHM is the logarithmic function with base 10. On the TI-83/84, look for the button. This is used to evaluate the common log (base 10) only. Example: Evaluate f(x)=log10x for x = 400. Round to four decimal places. Solution: f(400) = log Answer: ___________ LOG LOG ENTER

Antilog of the Common Log
We can also find a number given its logarithm. We say that N is the antilog of We use 2ND LOG [10x] Example: log N = N = _____________________________

Application of the Logarithm
Example Measured on the Richter , the magnitude of an earthquake of intensity I is defined to be R = Log(I/I0), where I0 is a minimum level for comparison. What is the Richter scale reading for the 1995 Philippine earthquake for which I=20,000,000 I0?

Section 13.5: The Natural Logarithmic Function
In section 13.1, we saw the natural exponential function with base e. Its inverse is the natural logarithmic function with base e. Instead of writing the natural log as logex, we use the notation ln x, which is read as “the natural log of x” and is understood to have base e.

The Natural Logarithmic Function
To evaluate the natural log using the TI-83/84, use the button. Example Evaluate the function f(x) = ln x at a) x = 1.5 b) x = -2.3 LN This means that ______________________________________

Graph of the Natural Exponential and Natural Logarithmic Function
f(x) = ex and g(x) = ln x are inverse functions and, as such, their graphs are reflections of one another in the line y = x.

Antilog of the Natural Logarithm
We say that N is the antilog of We use 2ND LOG [ex] Example: ln N = N = _____________________________ 2ND LN

Change-of-Base Formula
I mentioned that the calculator only has two types of log keys, the COMMON LOG (BASE 10) and the NATURAL LOG (BASE e). It’s true that these two types of logarithms are used most often, but sometimes we need to evaluate logarithms with bases other than 10 or e. To do this on the calculator, we use a CHANGE-OF-BASE FORMULA. We will convert the logarithm with base a into an equivalent expression involving common logarithms or natural logarithms.

Change-of-Base Formula (continued)
Let a, b, and x be positive real numbers such that a  1 and b  1. Then logbx can be converted to a different base using any of the following formulas.

Change-of-Base Formula Examples*
Use the change-of-base formula to evaluate log7264 using common logarithms using natural logarithms. Solution: The result is the same whether you use the common log or the natural log.

Change-of-Base Formula Examples
Use the change-of-base formula to evaluate a) b)

Graph of the Logarithmic Function with base b
MAT 205 SP 2009 Graph of the Logarithmic Function with base b a) Graph on the calculator. b) Graph its inverse on the calculator.

Section 13.3: Properties of Logarithms
MAT 205 SP 2009 Section 13.3: Properties of Logarithms Let b be a positive real number such that b  1, and let n, x, and y be real numbers. Base b Logarithms Natural Logarithms

WARNING!!!!!!

Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:

Again!  Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:

This is fun!  Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:

Try these!  Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:

Properties of Logarithms Rock! 
Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:

One more!  Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:

Solving EXPONENTIAL Equations: Part I
MAT 205 SP 2009 Section 13.6: Solving Exponential and Logarithmic Equations Solving EXPONENTIAL Equations: Part I I. Using the One-to-One Property If you can write the equation so that both sides are expressed as powers of the SAME BASE, you can use the property bx = by if and only if x = y. Example: Solve 4x-2 = 64

Solving EXPONENTIAL Equations: Part II
MAT 205 SP 2009 Solving EXPONENTIAL Equations: Part II II. By Taking the Logarithm of Each Side ISOLATE the exponential term on one side of the equation. TAKE THE COMMON OR NATURAL LOG of each side of the equation. USE THE PROPERTIES OF LOGARITHMS to remove the variable from the exponent. SOLVE for the variable. Use the calculator to evaluate the resulting log expression.

Solving EXPONENTIAL Equations
Example: Solve 3(54x+1) -7 = Give answer to 3 decimal places.

Solving LOGARITHMIC Equations: Part I
I. Using the One-to-One Property If you can write the equation so that both sides are expressed as SINGLE logarithms with the SAME BASE, you can use the property logbx = logby if and only if x = y.

Solving LOGARITHMIC Equations: Part I
Example of one-to-one property: Solve log3x + 2log35 = log3(x + 8)

Solving Logarithmic Equations: Part II
II. By Rewriting in Exponential Form USE THE PROPERTIES OF LOGARITHMS to combine log expressions into a SINGLE log expression, if necessary. ISOLATE the logarithmic expression on one side of the equation. Rewrite the equation in EXPONENTIAL FORM. SOLVE the resulting equation for the variable. CHECK the solution in the original equation either graphically or algebraically

Solving Logarithmic Equations
Example: Solve

Solving Exponential and Logarithmic Equations GRAPHICALLY
Remember, you can verify the solution of any one of these equations by finding the graphical solution using your TI-83/84 calculator. Enter the left hand side of the original equation as y1, Enter the right side as y2, and Find the point at which the graphs intersect. Below is the graphical solution for the last example. The x-coordinate of the intersection point is approximately , confirms our algebraic solution.

I. Solve each of the following EXPONENTIAL equations
I. Solve each of the following EXPONENTIAL equations. Round to 4 decimal places, if necessary.

MAT 205 SP 2009

MAT 205 SP 2009

Challenge Question

II. Solve each of the following LOGARITHMIC equations
II. Solve each of the following LOGARITHMIC equations. Round to 4 decimal places, if necessary.

MAT 205 SP 2009 Applications Example: How long will it take \$25,000 to grow to \$500,000 if it is invested at 9% annual interest compounded monthly? Round to the nearest tenth of a year. Formula:

Another Example: The population of Asymptopia was 6500 in 1985 and has been tripling every 12 years since then. If this rate continues, when will the population reach 75,000? Let t represent the number of years since 1985 P(t) represents the population after t years.

Drug medication: The formula can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug has been administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections? Round to the nearest tenth of an hour.

A Logarithmic Model: The loudness L, in bels (named after ?), of a sound of intensity I is defined to be where I0 is the minimum intensity detectable by the human ear. The bell is a large unit, so a subunit, the decibel, is generally used. For L, in decibels, the formula is

A Log Model (cont) Find the loudness, in decibels, for each sound with the given intensity. Library Dishwasher Loud muffler

A Log Model (cont) If the front rows of a rock concert has a loudness of 110 dB and normal conversation has a loudness of 60 dB , how many times greater is the intensity of the sound in the front rows of a rock concert than the intensity of the sound of normal conversation?

x y Section 13.7: Graphs on Log and Semi-log Paper
MAT 205 SP 2009 Section 13.7: Graphs on Log and Semi-log Paper Let’s say we want to plot the graph y = 5x x y -1 1 2 3 4 5 The detail for values of x less than 3 is nearly imperceptible.

MAT 205 SP 2009 Often times, we want to model data that require that small variations at one end of the scale are visible, while large variations at the other end are also visible. To graph functions where one or both of the variables have a wide change in values, we can use a logarithmic scale. This type of scale is marked off in distances that are proportional to the logarithm of the values being represented. The distances between integers on a log scale are not equal, but will give us a better way to show a greater range of values.

The horizontal scale has equal spacing between the lines
MAT 205 SP 2009 If we want to show a large range of values for only one of your variables, we will use SEMI-LOG paper. Semi-log paper has two scales: The horizontal scale has equal spacing between the lines The vertical scale does not have equal spacing between the lines. It uses a logarithmic scale.

MAT 205 SP 2009 Semi-log paper allows you to graph exponential data without having to translate your data into logarithms—the paper does it for you. The scale of semi-log paper has cycles. Below is what is known as 3-cycle semi-log graph paper. On the vertical scale, the powers of ten are evenly spaced. On the horizontal scale, the numbers along the axis are evenly spaced.

x y Let’s see the graph of y = 5x on semi-log paper. -1 0.2 1 5 2 25 3
MAT 205 SP 2009 Let’s see the graph of y = 5x on semi-log paper. x y -1 0.2 1 5 2 25 3 125 4 625 3125

MAT 205 SP 2009 Semi-log paper is often used to transform a nonlinear data relation into a linear one. If a function makes a STRAIGHT LINE when graphed on semi-log graph paper, we call it an EXPONENTIAL FUNCTION.

On log-log paper, both axes are marked with a logarithmic scale.
MAT 205 SP 2009 On log-log paper, both axes are marked with a logarithmic scale.

x y Example: Create a log-log plot of the function y = 0.5x3. 0.5 1 2
MAT 205 SP 2009 Example: Create a log-log plot of the function y = 0.5x3. x y 0.5 1 2 5

MAT 205 SP 2009 Notice that the graph of y = 0.5x3 on log-log paper is a straight line. An equation in the form of y = axb is called a POWER FUNCTION. If you plot the data points of a power function on log-log paper, it appears LINEAR.