# Chapter 3 Exponential & Logarithmic Chapter 3 Exponential & Logarithmic Functions Section 3.1...

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Chapter 3 Exponential & Logarithmic Functions

Section 3.1 Exponential Functions & Their Graphs

Definition of Exponential Function: The exponential function f with base a is denoted

f(x) = ax where a > 0, a≠ 1 and x is any real number. Example 1: Evaluating Exponential Expressions Use a calculator to evaluate each expression

a. 2-3.1

b. 2-π Example 2: Graphs of y = ax

In the same coordinate plane, sketch the graph of each function.

a. f(x) = 2x

b. g(x) = 4x

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Example 3: Graphs of y = a-x

In the same coordinate plane, sketch the graph of each function.

a. F(x) = 2-x

b. G(x) = 4-x Compare the functions in examples 2 and 3. F(x) = 2-x = and G(x) = 4-x =

Graph of y = ax

Graph of y = a-x

Domain

Range

Intercepts

Increasing/ Decreasing

Horizontal Asympotote

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Example 4: Sketching Graphs of Exponential Functions Compare the following graphs to g(x) = 3x

a. g(x) = 3x+1 b. h(x) = 3x – 2

c. k(x) = -3x d. j(x) = 3-x The Natural Base e e ≈ 2.71828... is called the natural base. The function f(x) = ex is called the natural exponential function. Example 5: Evaluating the Natural Exponential Function Use a calculator to evaluate each expression.

a. e-2

b. e-1

c. e1

d. e2

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Example 6: Graphing Natural Exponential Functions Sketch the graph of each natural exponential function.

a. f(x) = 2e0.24x b. g(x)= e-0.58x Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1) For n compoundings per year: A = P(1 + �/�)nt

2) For continuous compounding: A = Pert Example 7: Compounding n Times and Continuously A total of $12,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded

a) Quarterly

b) Continuously

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Example 8: Radioactive Decay In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread radioactive chemicals over hundreds of square miles, and the government evacuated the city and the surrounding area. To see why the city is now uninhabited, consider the following model: P = 10e-0.0000845t

This model represents the amount of plutonium that remains (from an initial amount of 10 pounds) after t years. Sketch the graph of this function over the interval from t = 0 to t = 100,000. How much of the 10 pounds will remain after 100,000 years?

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Section 3-2: Logarithmic Functions and Their Graphs

LOGARITHMIC FUNCTIONS Remember… From Section 1.6 – a function has an inverse if it passes the Horizontal Line Test (no

horizontal line intersects the graph more than once).

• Exponential functions of the form f (x) = ax

pass the Horizontal Line Test, which means they

must have an inverse.

• The inverse of an exponential function is called the logarithmic function with base a.

Try and start finding the inverse of: f (x) = a x

Where do you run into trouble?

DEFINITION OF LOGARITHMIC FUNCTION:

For x > 0 and 0 < a ≠1,

y = loga x if and only if x = ay .

The function given by

f (x) = loga x

is called the logarithmic function with base a.

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Remember… A logarithm is an exponent! This means that loga x is the exponent to which a must

be raised to obtain x. For instance, log2 8 = 3 because 2 must be raised to the third power to get 8.

Example 1: Evaluating Logarithms

a. log2 32 = d. log10 1

100 =

b. log3 27 = e. log31 =

c. log4 2 = f. log2 2 =

The logarithmic function with base 10 is called the common logarithmic function.

Example #2: Evaluating Logarithms on a Calculator Use a calculator to evaluate each expression.

a. log1010 b. 2log10 2.5 c. log10(−2)

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PROPERTIES OF LOGARITHMS:

1. loga 1 = 0 because…

2. loga a =1 because…

3. loga a x = x because…

4. If loga x = loga y , then x = y . Example #3: Using Properties of Logarithms Solve each equation for x. List which property you used.

a.) log2 x = log2 3

b.) log4 4 = x

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GRAPHS OF LOGARITHMIC FUNCTIONS

Remember… The graphs of inverse functions are reflections of each other in the line y = x .

Example #4: Graphs of Exponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function.

a. f (x) = 2x

b. g(x) = log2 x Example #5: Sketching the Graph of a Logarithmic Function

Sketch the graph of the common logarithmic function f (x) = log10 x .

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Basic Characteristics of Logarithmic Graphs:

Graph of y = loga x , a >1

• Domain:

• Range:

• Intercept:

• Increasing or decreasing?

• Vertical asymptote:

• Reflection:

Example #6: Sketching the Graphs of Logarithmic Functions

Sketch each of the following, referring to f (x) = log10 x . Be sure to describe the transformation.

a. g(x) = log10(x −1)

b. h(x) = 2 + log10 x

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THE NATURAL LOGARITHMIC FUNCTION The logarithmic function with base e is the natural logarithmic function and is denoted by the special

symbol ln x , read as “el en of x.”

PROPERTIES OF NATURAL LOGARITHMS:

1. ln1 = 0 because…

2. lne =1 because…

3. lnex = x because…

4. If ln x = ln y , then x = y .

Example #7: Using Properties of Natural Logarithms

a. ln 1 e

= c. lne0 =

b. lne2 = d. 2lne = Example #8: Evaluating the Natural Logarithmic Function Use a calculator to evaluate each expression.

a. ln2 b. ln0.3 c. lne2 d. ln(−1)

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Example #9: Finding the Domain of Logarithmic Functions Find the domain of each function.

a. f (x) = ln(x − 2)

b. g(x) = ln(2 − x)

c. h(x) = ln x 2 Example #10: Human Memory Model Students participating in a psychological experiment attended several lectures on a subject and were

given an exam. Every month for a year after the exam, the students were retested to see how much of

the material they remembered. The average scores for the group are given by the human memory

model

f (t) = 75 − 6ln(t +1) , 0 ≤ t ≤12 where t is the time in months.

a. What was the average score on the original (t = 0) exam?

b. What was the average score at the end of t = 2 months?

c. What was the average score at the end of t = 6 months?

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Section 3-3 Properties of Logarithms You know that you can only put ln or log10 into your calculator. In order to evaluate other logs, you will need the change-of-base formula. Change-of-Base Formula Let a, b, and x be positive real numbers such that a≠1 and b≠1. Then loga x is given by

loga x = �����

����

Example 1: Changing Bases Using Common Logarithms

a. log430 =

b. log214 = Example 2: Changing Bases Using Natural Logarithms

a. log430 =

b. log214 = Properties of Logarithms You know from the previous section that the logarithmic function is the inverse of the exponential function. Thus, it makes sense that properties of exponents should have corresponding properties involving logarithms. Properties of Logarithms Let a be a positive number such that a ≠1, and let n be a real number. If u and v are positive real numbers, the following properties are true:

1. logauv = logau + logav 1. ln (uv) = ln u + ln v 2. loga

� = logau - logav 2. ln

� = ln u – ln v

3. logaun = nlogau 3. ln un = n ln u

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Example 3: Using Properties of Logarithms Write the logarithm in terms of ln 2 and ln 3

a. ln 6 b. ln �

�

Example 4: Using Properties of Logarithms Use the properties of logarithms to verify that – ln ½ = ln 2 Rewriting Logarithmic Expressions Example 5: Rewriting the Logarithm of a Product log105x3y Example 6: Rewriting the Logarithm of a Quotient

ln √����

Example 7: Condensing a Logarithmic Expression �

� log10x + 3log10(x+1)

Example 8: Condensing a Logarithmic Expression 2ln (x+2) – ln x

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