ELF.01.9 - Solving Logarithmic Equations

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ELF.01.9 - Solving Logarithmic Equations

• MCB4U - Santowski

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(A) Introduction to Logarithmic Equations

• Many measurement scales used for naturally
  occurring events like earthquakes, sound
  intensity, and acidity make use of logarithms
• Additionally, since we know how logarithms and
  exponents are related, many applications
  involving exponents can also be mathematically
  analyzed using logarithms
• So in working with these types of problems, we
  need to know how to solve logarithmic equations

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(B) Example

• The magnitude, R, of an earthquake on the Richter
  scale is given by the equation R log(a/T) B,
  where a is the amplitude of the vertical ground
  motion (measured in microns), T is the period of
  the seismic wave (measured in seconds) and B is a
  factor that accounts for the weakening of the
  seismic waves. Find the amplitude of the vertical
  ground motion for an earthquake that measured 6.3
  on the Richter scale, and the period of the
  seismic wave was 1.6 seconds and B 4.2

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(B) Example

• Givens
• R log(a/T) B
• R 6.3
• T 1.6 sec
• B 4.2
• 6.3 log(a/1.6) 4.2
• 2.1 log(a/1.6)
• ? now recall converting logs to exponents
• 10(2.1) a/1.6
• 125.89 a/1.6
• 201.4 microns a
• Therefore, the amplitude of the ground wave
  would be approx 201 microns

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(C) Solving Simple Log Equations

• Solve the following equations

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(D) Internet Links to Simple Equations

• Now work through the following worksheet to
  reinforce your skills with simple logarithms
• From edHelper.com - Logarithms

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(E) Strategies for Solving Harder Logarithmic
Equations

• the two key ideas in solving logarithmic
  equations are
• (1) get both sides of the equation to be a single
  logarithmic expression
• (2) both sides must have logarithms
• the key tools you will use to work through
  logarithmic equations are the three laws -
  product law, quotient law and power law.

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(E) Examples

• Solve and verify and state the restrictions for
  log3(x) - log3(4) log3(12)
• Solve and verify and state the restrictions for
  log6(x 3) log6(x - 2) 1
• Solve and verify and state the restrictions for
  log2(3x 2) 3 - log2(x - 1)

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(F) Connections to Graphs

• We will work out the following examples on a GDC
• Graph y1 log(x) log(4)
• Graph y2 log(12)
• Find their intersection
• Which occurs when x 48

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(F) Connections to Graphs

• But what happens when the base of the logarithm
  is different??? The GDC works in base 10!!!!
• We will use a different base (e) ? ln (loge) on a
  GDC
• Graph y1 ln(x) ln(4)
• Graph y2 ln(12)
• Find their intersection .
• Which occurs when x 48 (although the y-value
  has changed) ? so base does NOT matter for the
  solution of x

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(F) Connections to Graphs

• Recall that solving equations also relates to
  finding x-intercepts ? how does that apply here?
• If we rearrange the equation, we get ?
• log(x) log(4) log(12) 0
• Now graph this rearranged equation and find the
  x-intercept(s)

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(G) Internet Links

• Solving Logarithmic Equations Lesson - From
  PurpleMath
• SOLVING LOGARITHMIC EQUATIONS - from SOSMath
• College Algebra Tutorial on Logarithmic Equations
  from WTAMU

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(D) Homework

• Nelson text, page 146, Q3,6,7,8,12,16-19