Chapter 8 Notes



Chapter 8 Notes Name ____________________________________

8.1 Exponential Growth

• Exponential Function: expression [pic] where the base [pic].

• Asymptote: is a __________ that a graph approaches as you move ___________ from the origin (the graph will never hit it) – dashed line.

• Exponential Growth Function: if [pic], the function [pic].

Ex. 1 Graph and state the domain and range.

A) [pic]

B) [pic]

• To graph: [pic]

▪ Graph [pic]

Then move each point:

o ___ units _______________________ [pic]

o ___ units _______________________ [pic]

NOTE:

✓ Asymptote : [pic]

✓ Domain: [pic]

✓ Range: [pic]

Ex. 2 Graph [pic]. State the domain and range.

* When a real-life quantity ___________________ by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by:

[pic]

Where

[pic]

Ex. 3 In 1990 the cost of tuition at a state university was $4300. During the next 8 years, the tuition rose 4% each year.

A) Write a model that gives the tuition y (in dollars) t years after 1990.

B) Graph the model (go from 0 – 8 years)

* Compound Interest = consider an initial principal P deposited in an account that pays interest at an annual rate r (decimal) compounded ____ times per year. The amount A in the account after t years is:

[pic]

Ex. 4 You deposit $1500 in an account that pays 6% annual interest. Find the balance after 1 year if the interest is compounded:

A) Annually

B) Semiannually

C) Quarterly

8.2 Exponential Decay

• Exponential Decay Function: if [pic], the function [pic].

Ex. 1 State whether [pic]is an exponential growth or an exponential decay function.

A) [pic] B) [pic]

Ex. 2 Graph and state the domain and range.

A) [pic]

B) [pic]

• To graph: [pic]

▪ Graph [pic]

Then move each point:

o ___ units horizontally [pic]

o ___ units vertically [pic]

NOTE:

✓ Asymptote : [pic]

✓ Domain: [pic]

✓ Range: [pic]

Ex. 3 Graph [pic]. State the domain and range.

* When a real-life quantity ___________________ by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by:

[pic]

Where

[pic]

Ex. 4 There are 40,000 homes in your city. Each year 10% of the homes are expected to disconnect from septic systems and connect to the sewer system.

A) Write an exponential decay model for the number of homes that still use septic systems. Estimate the number of homes after 5 years.

B) Graph the model and estimate when about 17,200 homes will still not be connected to the sewer system.

8.3 The Number e

• The Natural Base e: (Euler’s number)

o Irrational

o As n approaches [pic],

▪ [pic] approaches [pic]

Ex. 1 Simplify

A) [pic] B) [pic] C) [pic] D) [pic]

Ex. 2 Evaluate with a calculator

A) [pic] B) [pic] C) [pic]

• Natural Base Exponential Function:

[pic]

o [pic]

o [pic]

• To graph: [pic]

▪ Graph [pic]

Then move each point:

o ___ units horizontally [pic]

o ___ units vertically [pic]

NOTE:

✓ Asymptote : [pic]

✓ Domain: [pic]

✓ Range: [pic]

Ex. 3 Graph the function. State the domain and range.

A) [pic]

B) [pic]

* From 8.1, the function for compound interest is [pic]. As n approaches [pic], then

• Continuously Compound Interest:

[pic]

Ex. 4 You deposit $1500 in an account that pays 7.5% annual interest compounded continuously. What is the balance after 1 year?

8.4 Logarithmic Functions

* [pic]

Since,

[pic]

To find x: we use logarithms.

* Logarithm with Base b: [pic] (logarithm of y with base b is denoted):

Logarithmic Form Exponential Form

“Log base b of y”

[pic]

Ex. 1 Write the logarithmic equation in exponential form.

A) [pic] B) [pic] C) [pic]

* Special Logarithm values: b is positive where [pic]:

• Logarithm of 1: [pic]

• Logarithm of base b: [pic]

Ex. 2 Evaluate the expression

A) [pic] B) [pic]

C) [pic] D) [pic]

Note:

* If you need to root it: answer is a ____________________

* If you need or start with a decimal/fraction: answer is ___________________

* Common Logarithm = base _____

[pic]

* Natural Logarithm = base _____

[pic]

Ex. 3 Evaluate with a calculator

A) [pic] B) [pic]

* The logarithmic function [pic] is the inverse of the exponential function [pic]. So,

[pic]

they “undo” each other.

Ex. 4 Simplify

A) [pic] B) [pic] C) [pic] D) [pic]

Ex. 5 Find the inverse of the function (recall: switch ___ and ___…then solve for ___)

A) [pic] B) [pic] C) [pic]

• Graphs of Logarithmic Functions:

o Graph of [pic]

▪ Graph [pic] and shift ___ units [pic] and ___ units [pic].

▪ The line [pic] is a vertical asymptote (graph can’t touch this line).

▪ The domain is [pic].

▪ The range is [pic].

• If [pic], graph moves _____ to the right.

• If [pic], graph moves __________ to the right.

Ex. 6 Graph and state the domain and range.

A) [pic]

B) [pic]

C) [pic]

8.5 Properties of Logarithms

• Properties of Logarithms: Let b, u, and v be positive numbers such that[pic],

o Product Property:

[pic]

o Quotient Property:

[pic]

o Power Property:

[pic]

Ex. 1 Use [pic] to approximate:

A) [pic]

B) [pic]

C) [pic]

Ex. 2 Expand (assume x is positive)

A) [pic]

B) [pic]

Ex. 3 Condense (assume x is positive)

A) [pic]

B) [pic]

• Change-Of-Base Formula: Let u, b, and c be positive with [pic] (since your calculator only has base 10…this is so you can use the calculator for any base).

[pic] & [pic]

Ex. 4 Evaluate by change-of base

A) [pic]

B) [pic]

C) [pic]

8.6 Solving Exponential and Logarithmic Equations

* For [pic], [pic].

* For [pic], [pic]

• Since the ________________ of log functions is usually not all real numbers, you need to check for extraneous solutions.

Solve

Ex. 1

A) [pic] B) [pic]

Ex. 2 [pic]

Ex. 3 [pic]

Ex. 4 [pic]

Ex. 5

A) [pic] B) [pic]

Ex. 6 [pic]

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[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

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