Chapter 8: Exponential and Logarithmic Functions

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Chapter 8 Exponential and Logarithmic Functions

• A What You Need To Know Guide

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Exponential Functions

• Basic equation y abx
• Growth functions
• Decay functions
• Half-life functions
• Interest problems
• Word problems

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Growth Decay y abx

• r is rate of change, typically given as a
  percent. (Must be changed to a decimal!)
• Growth
• r rate of increase
• b 1 r
• Population increase, value appreciation, cell
  reproduction, etc.

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Growth Decay y abx

• Decay (0 lt b lt 1)
• r of decrease can be given as a negative
  number
• b 1 r
• Value depreciation, medication metabolization,
  half-life
• Half-Life
• Special type of decay question
• b 1 r where r is always 50, making b ½
• Decay of materials (Carbon-14, Technecium-99m,
  etc)

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Growth Decay General

• 1. Set-up y abx
• Total value (initial value)(growth/decay
  factor)(time)
• 2. Look (Graph)

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Graphing Exponentials

• To graph
• Create an xy chart
• Use the values x -2, -1, 0, 1, 2 for x-values
• Evaluate to find corresponding y-values
• Graph the points on the coordinate grid
• Sketch the graph
• Growth functions y-values get bigger from -2 to
  2 (go up from left to right)
• Decay functions y-values get smaller from -2 to
  2 (go down from left to right)

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Growth Decay . . .

• are EXPONENTIAL FUNCTIONS (they contain
  exponents).
• Are evaluated using PEMDAS
• ARE NOT SOLVED USING LOGS! (they are evaluated!)
• are graphed using xy charts, plotting points,
  and sketching the curve

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Growth Decay Word Problems
Growth
Formula y a (1 r)x
Key Words Increase, Growth, Appreciate, etc.
Important Info The rate of change, b (or 1 r), is greater than 1. Answer should be larger than the initial value, a.
Number Placement End amount (initial amount)(rate of change)(time)
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Growth Decay Word Problems
Decay
Formula y a (1 - r)x
Key Words Decrease, depreciate, decline, etc.
Important Info The rate of change, b (or 1 - r), has a value between 0 and 1, exclusively. Answer should be smaller than the initial value, a.
Number Placement End amount (initial amount)(rate of change)(time)
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Growth Decay Word Problems
Half Life
Formula y a (1/2)x or y a(0.5)x Exponent is (time elapsed/length of half-life)
Key Words Half-life, decreases by half, etc.
Important Info The rate of change, b, always has a value of ½. Answer should be smaller than the initial value, a.
Number Placement End amount (initial amount)(1/2)(time elapsed/length of half-life)
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Calculator Tips

• ALWAYS turn percents into decimals before using
  them to evaluate ANYTHING!
• ALWAYS follow the order of operations when
  evaluating (PEMDAS)
• Input only one operation at a time
• ALWAYS put the exponent for half-life in
  (parenthesis)!!!

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Logarithms

• Are the inverses of exponentials (they undo
  exponentials).
• Allow you to solve an equation with a variable in
  the exponent.
• Are graphed using the exponential version of the
  equation
• Follow VERY similar rules to exponents

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Graphing Logarithms

• To graph
• Switch the logarithm into its exponential
  counterpart.
• Create an xy chart.
• Use the values x -2, -1, 0, 1, 2 for x-values.
• Evaluate to find corresponding y-values.
• To get the log graph, switch the x and y values.
• Graph the points and sketch the graph.

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Logarithms as Inverses

• The logarithm function is the inverse of the
  exponential function.
• Definition of a logarithm (use this to change
  forms)
• logby x if y bx
• Iog10 called a common logarithm is the inverse
  of a base 10 exponential function
• LN or ln is called a natural logarithm and is
  the inverse of a base e exponential function.

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Evaluating a Logarithm

• For a base 10 logarithm, use the log button on
  your calculator
• For a base e logarithm, or a natural logarithm,
  use the ln button on your calculator
• For all other bases, use the change of base
  formula
• logbM log M
• log b

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Properties of Logarithms

• VERY similar to exponential properties
• Used to combine like terms for logarithms (in
  log equations)
• Product Property
• logbM logbN logbMN
• Quotient Property
• logbM logbN logb(M/N)
• Power Property
• logbMx x logbM

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Solving Exponential Equations

• Steps
• Isolate the exponential term (bx) using the
  reverse order of operations.
• Take the common logarithm on both sides (unless
  its a base e, in that case you should use the
  natural log LN)
• Solve the remaining equation for the given
  variable using the log properties and reverse
  PEMDAS

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Solving Exponential Equations

• Example
• Solve 4 5 72x-1
• Example using e
• Solve 4e2x1 5 12

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Solving Logarithmic Equations

• Steps
• Use the log properties to combine like terms,
  getting one logarithm, isolated.
• Raise both sides to a power position, using the
  base that matches the given logarithm (LN is base
  e!!) to cancel the log.
• Use reverse PEMDAS to solve for the given
  variable.

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Solving Logarithmic Equations

• Example
• Solve log(2x 1) 10
• Solve log2 log 3x 7
• Solve log3x-1 9
• Example using LN
• Solve LN3x 5 11
• 2LNx 2 15