6.6 Logarithmic and Exponential Equations

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6.6 Logarithmic and Exponential Equations

• In this section, we will study the following
  topics
• Solving logarithmic equations
• Solving exponential equations
• Using exponential and logarithmic equations to
  solve real-life problems.

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Strategies for Solving Logarithmic Equations

• There are two basic strategies for solving
  logarithmic equations
• Converting the log equation into an exponential
  equation by using the definition of a
  logarithm
• Using the One-to-One Property

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Solving Logarithmic Equations

• I. Converting to Exponential Form
• ISOLATE the logarithmic expression on one side of
  the equation.
• CONVERT TO EXPONENTIAL FORM
• SOLVE for x. Give approximate answers to 3
  decimal places, unless otherwise indicated.

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Example Converting to Exponential Form
Solve
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Solving Logarithmic Equations

• II. Using the One-to-One Properties
• Can be used if the equation can be written so
  that both sides are expressed as SINGLE
  LOGARITHMS with the SAME BASE.
• Use the properties of logarithms to CONDENSE the
  log expressions on either side of the equation
  into SINGLE LOG expressions.
• Apply the ONE-TO-ONE PROPERTY.
• SOLVE for x. Give approximate answers to 3
  decimal places, unless otherwise indicated.

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Example One-to-One Property for Logs
Solve log3x 2log35 log3(x 8)
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PRACTICE!!
Solve each of the following LOGARITHMIC
equations.
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PRACTICE!!
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PRACTICE!!
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PRACTICE!!
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Strategies for Solving Exponential Equations

• There are two basic strategies for solving
  exponential equations
• Using the One-to-One Property
• Taking the natural or common log of each side.

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Solving Exponential Equations

• Using the One-to-One property
• Can be used if the equation can be written so
  that both sides are expressed as powers of the
  SAME BASE
• Use the properties of exponents to CONDENSE the
  exponential expressions on either side of the
  equation into SINGLE exponential expressions.
• Apply the ONE-TO-ONE PROPERTY.
• SOLVE for x. Give approximate answers to 3
  decimal places, unless otherwise indicated.

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Example One-to-One Property
Solve
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Example One-to-One Property
Solve
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Solving Exponential Equations

• Taking the Natural or Common Log of Each Side
• If there is one exponential term, ISOLATE the
  exponential term on one side of the equation.
• TAKE THE NATURAL OR COMMON LOG OF EACH SIDE of
  the equation.
• Use the property to
  get the variable out of the exponent.
• SOLVE for x. Give approximate answers to 3
  decimal places, unless otherwise indicated.

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Example Taking the Log of Each Side
Solve 3(54x1) -7 10
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Example Taking the Log of Each Side
Solve.
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PRACTICE!!
Solve each of the following EXPONENTIAL
equations.
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PRACTICE!!
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PRACTICE!!
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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
  in y1
• Enter the right side in y2
• Find the point at which the graphs intersect.
  Below is the graphical solution of the equation
  2ln(x - 5) 6

The x-coordinate of the intersection point is
approximately 25.086. This is the (approx)
solution of the equation.
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Exponential Growth Example
The population of Asymptopia was 6500 in 1970 and
has been tripling every 12 years since then. In
what year will the population reach 75,000?
Let t represent the number of years since 1970
P(t) represents the population after t years.
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A Compound Interest Example
How long will it take 25,000 to grow to 500,000
at 9 annual interest compounded continuously?
Use the compound interest formula
Where P Principal (original amount invested or
borrowed) r annual interest rate t
number of years money is invested A the
amount of the investment after t years
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A Compound Interest Example (cont.)
Substitute in the given values and solve for t.
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• End of Section 6.6