Exponents and Logarithms

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Exponents and Logarithms

• Equations

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Learning Goal

• I will be able to solve logarithmic and
  exponential equations.

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Laws of Logarithms
loga rs loga r loga s
loga r/s loga r - loga s
loga rc c loga r
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Laws of Logarithms
Use laws of logarithms to express as a sum,
difference, or product of simple logarithms.
ln(xy3) ln x ln y3 ln x 3 ln y
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Things to think about when solving equations
Carefully note the reasons why we move from step
to step. Every logarithmic or exponential
equation is unique so theres no one rule for
what to do! Often, using the laws of logs or the
laws of exponents will help. If all else fails,
use technology. (But only after algebra fails)
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Facts to keep in mind
Bases must be positive Arguments must be positive
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Solve Example 1
3x1 81 log3 81 x 1 4 x1 x 3
3x1 81 3x1 34 x 1 4 x 3
OR
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Solve Example 2
Is this valid?
NO
Arguments must be positive
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Solve Example 3
Is this valid?
YES
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Solve - Example 4
3 ln 5x 2 16 3 ln 5x 2 16 3 ln 5x 18 ln
5x 6 e6 5x
A fine answer
An okay approximation
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Solve - Example 6
½ log5 (x 6) log5 x 0 ½ log5 (x 6) log5
x log5 (x 6)1/2 log5 x (x 6)1/2 x x 6
x2 x2 x 6 0 (x 3)(x 2) 0 x 3, -2 x
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or log5 (x 6)1/2 log5 x 0 log5 (x
6)1/2/x 0
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Solve Example 7
2x 5 log2 5 x
x 2.322
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Solve Example 8
5x - 2 3 3x 2 log 5x-2 log 33x2 (x-2) log
5 (3x2) log 3 x log 5 2 log 5 3x log 3 2
log 3 x log 5 3x log 3 2 log 3 2 log
5 x(log 5 3 log 3) 2 log 3 2 log 5
x -3.212
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Learning Goal

• I will be able to solve logarithmic and
  exponential equations.

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End of notes.

• Now do this Gateway Problem
• Algebraically solve ln (3x 5) 8.

A Gateway Problem is an opportunity to see if you
have mastered the learning goal. You should not
discuss the gateway problem with anyone while
doing it we need to see what you know.
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Gateway Problem Answerx e8/3 5/3

• If successful, homework is p. 213 All Vocab
  Check and 3, 7, 23, 31, 45, 51, 55, 77, 89, 111,
  119. If not successful, homework is p. 213 All
  Vocab Check and 3, 7, 17, 23, 27, 31, 35, 39, 45,
  51, 55, 73, 77, 87, 89, 111, 119.