Exponential and Logarithmic FUNctions

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Pre CalcChapter 4

• Exponential and Logarithmic FUNctions

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Exponential Functions4.1
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Exponential Functions

• For , the exponential function with
  a base a is defined byFor , the domain of
  the f is , the range of f is
  ,and the graph of f has one of the following
  shapes

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Graphs of Exponentials
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Graphs of Exponentials
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Graphs of Exponentials
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Graphs of Exponentials
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The Natural Exponential Function

• The natural exponential function is the
  exponential function
• with base e. It is often referred to as the
  exponential function

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What is e?

• Eulers number
• First to call it exponential, hence the e
• Naturally occurs in many instances
• Spreading of disease
• Interest earned, etc.
• e button on your calculator?

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Values of e

• Evaluate the following

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Models

• An infectious disease spreads around a city of
  population 10,000. After t days, the numbers of
  persons who have the virus can be modeled by
• How many infected people are there initially?
• How many infected people are there after 1 day?
  2days? 5 days?
• Describe the behavior of the function

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Compound Interest


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How is interest figured?

• Interest rates are stated in annual amounts
• However, interest is compounded n times per year
• Soour new interest becomes

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FinallyCompound Interest
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Investing

• You look to invest 10,000 at a rate of 9 for 4
  years.
• How much is the account worth when compounding
• Annually
• Semiannually
• Quarterly
• Monthly
• Daily
• Continuously???

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Continuously Compounded Interest

• Remember our Euler model..
• You look to invest 10,000 at a rate of 9 for 4
  years

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p.343 1-4, 13-18, 43,46,49,52-54
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Logarithmic Functions4.2
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Log Functions

• Let a be some positive number not equal to 1.
  The log function of base a, denoted
  , is defined by
• is the exponent to which the
  base a must be raised to get x

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Log Functions

• Allow us to switch back and forth between log
  form and exponential form
• Examples

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Log Functions

• Allow us to switch back and forth between log
  form and exponential form
• Examples

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Log Functions

• Allow us to switch back and forth between log
  form and exponential form
• Examples

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Log Functions

• Allow us to switch back and forth between log
  form and exponential form
• Examples

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Graphs of logs

• We know logs and exponentials are inverses
• So what???

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Graphing

• How did we rewrite equations to find inverses?
• Switch x and y values
• Sowhen graphing inversesthe output is the input
• Vice versa

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Graphing
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Graphing
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Graphing
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Graphing
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Properties of Logarithms

• Raise a to the power 0 to get 1
• Raise a to the power 1 to get a
• Raise a to the power x to get
• is the power to which a must be
  raised to get x

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Common Logarithm

• Common Log since we use base 10 system
• This is on your calculator!

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Natural Logarithm

• The log with base e is called the natural log
• The inverse of natural log

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Properties of Natural Logs

• Exactly the same as regular logs!

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p.3561-29 Odd
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Laws of Logarithms4.3
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Laws of




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Rewriting as Separate Functions
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Examples of
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Examples of
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Examples of
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Combining into 1 equation
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Examples of
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Examples of
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Examples of
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Examples of
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Careful, careful, careful

• Although the Laws of tell us how
  to compute logarithms of products or quotients
• There is no rule for the log of a sum or
  difference!

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Change of Base Formula
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How is this helpful?

• Allows us to calculate the log of any numbers or
  any base
• Evaluate the following

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p.363 1-4, 11-14, 27-30, 39-42
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Exponential and Log Equations!4.4
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Guidelines for Solving

• Isolate the exponential expression on 1 side
• Take the log of each side
• Use the Laws to bring down the exponent
• Solve!
• Smile

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Solving
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Solving
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Solving
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Solving
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Applying

• You have 4000 to invest in a college fund. You
  are quoted at receiving a quarterly compounded
  interest rate of 6. You figure you need 7500
  from this account for school. How long until this
  account will reach your goal?

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p.3721-13, 19-21, 57-58
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Exponential and Log EquationsDay 2
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Solving Exponential Equations
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Solving Exponential Equations
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Solving Exponential Equations
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Solving Exponential Equations
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Modeling Exponential and Logarithmic Functions4.5
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Modeling

• Exponentials can be used to measure many things
• Population growth
• Radioactive decay
• Heat diffusion
• Etc.

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Exponential Growth Model

• Growth of a population over time t can be
  modeled
• Hmmm..is there another model weve used that
  looks like this?

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Predicting Population

• A population of deer have a yearly growth rate of
  3. The initial population is at 1500 deer.
• How many deer will there be in 10 years?

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Predicting Population

• A population of deer have a yearly growth rate of
  3. The initial population is at 1500 deer.
• How long until there are 3000 deer?

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