1. ELF.01.9 - Solving Logarithmic Equations MCB4U - Santowski

2. (A) Introduction to Logarithmic Equations Many measurement scales used for naturally occurring events like earthquakes, sound intensity, and acidity make use of logarithms 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 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 So in working with these types of problems, we need to know how to solve logarithmic equations

3. (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 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

4. (B) Example Givens: Givens: R = log(a/T) + B R = log(a/T) + B R = 6.3 R = 6.3 T = 1.6 sec T = 1.6 sec B = 4.2 B = 4.2 6.3 = log(a/1.6) + 4.2 6.3 = log(a/1.6) + 4.2 2.1 = log(a/1.6) 2.1 = log(a/1.6)  now recall converting logs to exponents  now recall converting logs to exponents 10 (2.1) = a/1.6 10 (2.1) = a/1.6 125.89 = a/1.6 125.89 = a/1.6 201.4 microns = a 201.4 microns = a Therefore, the amplitude of the ground wave would be approx 201 microns Therefore, the amplitude of the ground wave would be approx 201 microns

5. (C) Solving Simple Log Equations Solve the following equations: Solve the following equations:

6. (D) Internet Links to Simple Equations Now work through the following worksheet to reinforce your skills with simple logarithms: Now work through the following worksheet to reinforce your skills with simple logarithms: From edHelper.com - Logarithms From edHelper.com - Logarithms From edHelper.com - Logarithms From edHelper.com - Logarithms

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

8. (E) Examples Solve and verify and state the restrictions for log 3 (x) - log 3 (4) = log 3 (12) Solve and verify and state the restrictions for log 3 (x) - log 3 (4) = log 3 (12) Solve and verify and state the restrictions for log 6 (x + 3) + log 6 (x - 2) = 1 Solve and verify and state the restrictions for log 6 (x + 3) + log 6 (x - 2) = 1 Solve and verify and state the restrictions for log 2 (3x + 2) = 3 - log 2 (x - 1) Solve and verify and state the restrictions for log 2 (3x + 2) = 3 - log 2 (x - 1)

9. (F) Connections to Graphs We will work out the following examples on a GDC: We will work out the following examples on a GDC: Graph y 1 = log(x) – log(4) Graph y 1 = log(x) – log(4) Graph y 2 = log(12) Graph y 2 = log(12) Find their intersection Find their intersection Which occurs when x = 48 Which occurs when x = 48

10. (F) Connections to Graphs But what happens when the base of the logarithm is different??? The GDC works in base 10!!!! 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 (log e ) on a GDC: We will use a different base (e)  ln (log e ) on a GDC: Graph y 1 = ln(x) – ln(4) Graph y 1 = ln(x) – ln(4) Graph y 2 = ln(12) Graph y 2 = ln(12) Find their intersection …. Find their intersection …. Which occurs when x = 48 (although the y-value has changed)  so base does NOT matter for the solution of x Which occurs when x = 48 (although the y-value has changed)  so base does NOT matter for the solution of x

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

12. (G) Internet Links Solving Logarithmic Equations Lesson - From PurpleMath Solving Logarithmic Equations Lesson - From PurpleMath Solving Logarithmic Equations Lesson - From PurpleMath Solving Logarithmic Equations Lesson - From PurpleMath SOLVING LOGARITHMIC EQUATIONS - from SOSMath SOLVING LOGARITHMIC EQUATIONS - from SOSMath SOLVING LOGARITHMIC EQUATIONS - from SOSMath SOLVING LOGARITHMIC EQUATIONS - from SOSMath College Algebra Tutorial on Logarithmic Equations from WTAMU College Algebra Tutorial on Logarithmic Equations from WTAMU College Algebra Tutorial on Logarithmic Equations from WTAMU College Algebra Tutorial on Logarithmic Equations from WTAMU

13. (D) Homework Nelson text, page 146, Q3,6,7,8,12,16-19 Nelson text, page 146, Q3,6,7,8,12,16-19