Warm Up Simplify expressions with exponents Reminders Quiz

Reminders • Quiz next class on chapter 5: questions 1 -4 • Simplify expressions

Writing equations review Write a function to represent the amount after t units of

Local or Richter magnitude • If seismograph not 100 km from epicenter: – ML

Energy of Earthquakes • Energy that goes into an earthquake is released from the

Earthquake energy comparison (No) Lightning bolt Tornado Mt St. Helens

Moment Magnitude • MW = 2/3(log M 0) - 6. 0 where • M

p. H • The p. H scale changes very small values into manageable numbers:

Laws of Logarithms If M and N are positive real numbers and b is

Pause to start on section 5 -6 HW: • Pg. 200, 3 -33 multiples

Section 5. 7 Exponential Equations Changing Bases To solve exponential equations and to change

Example 1 In 2015, there about 7. 3 billion people in the world. If

In 2015, there about 7. 3 billion people in the world. If the population

Example 2 Suppose you invest P dollars at an annual rate of 6% compounded

Suppose you invest P dollars at an annual rate of 6% compounded continuously. How

Additional Example An investment of $500 is made at 3. 6% annual interest. How

The Change of Base Formula The change of base formula can be used to

HW: • Section 5 -6, Pg. 200, 3 -33 multiples of 3 • and

Slides: 34

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Warm Up

Simplify expressions with exponents •

Reminders • Quiz next class on chapter 5: questions 1 -4 • Simplify expressions with exponents • Write exponential equations • Test: Thursday after Thanksgiving (our 2 nd class of the week) • All of chapter 5, exponents and logarithms

Writing equations review Write a function to represent the amount after t units of time for each situation. 1. 23 grams of a compound with a half-life of 2500 years 2. 180 bacteria that quadruple themselves every 2 hours 3. A new car worth $55, 000 that depreciates 25% per year 4. A $175, 000 student loan with a 3. 97% annual interest rate

Phenomena modelled with logarithms

Local or Richter magnitude • If seismograph not 100 km from epicenter: – ML = log 10 (A) + C(distance) where • A is the maximum seismic wave amplitude in microns (10 -6 m) recorded on a standard seismograph • C is a correction factor that is a function of distance from the seismograph to the epicenter surface P S A

Energy of Earthquakes • Energy that goes into an earthquake is released from the elastic crust – Like a spring • Energy that comes out of an earthquake distributed between – Radiated (wave) energy • Motion – Breaking rocks – Frictional heating Hard to Measure

Earthquake energy comparison (No) Lightning bolt Tornado Mt St. Helens

Moment Magnitude • MW = 2/3(log M 0) - 6. 0 where • M 0 is seismic moment in Newton-meters. • Is now replacing other magnitude scales, such as Richter magnitude or surface wave magnitude. – Provides a consistent measure of size of earthquakes from the smallest microearthquakes to the greatest earthquakes ever recorded.

Loudness

decibels •

p. H

p. H • The p. H scale changes very small values into manageable numbers: • p. H=-log(hydrogen ion concentration) • (concentration in moles per liter) • More hydrogen ions= more acidic

• Warm Up

Laws of Logarithms If M and N are positive real numbers and b is a positive number other than 1, then: Law 1:

Laws of Logarithms If M and N are positive real numbers and b is a positive number other than 1, then: Law 2:

Laws of Logarithms If M and N are positive real numbers and b is a positive number other than 1, then: Law 3:

Laws of Logarithms If M and N are positive real numbers and b is a positive number other than 1, then: Law 4:

5. 6 Laws of Logarithms

For the next examples let: •

Pause to start on section 5 -6 HW: • Pg. 200, 3 -33 multiples of 3

Section 5. 7 Exponential Equations Changing Bases To solve exponential equations and to change the base of logarithms.

Example 1 In 2015, there about 7. 3 billion people in the world. If the population grows at 1. 95% per year, estimate the year when the population will be 10 billion people.

In 2015, there about 7. 3 billion people in the world. If the population grows at 1. 95% per year, estimate the year when the population will be 10 billion people. *We need to find the exponent!

Example 2 Suppose you invest P dollars at an annual rate of 6% compounded continuously. How long does it take: • To increase your investment by 50%? • To double your money?

Suppose you invest P dollars at an annual rate of 6% compounded continuously. How long does it take: • To increase your investment by 50%? *Divide both sides by P *ln both sides *Apply Law 4 *lne = 1 *Solve for t

Suppose you invest P dollars at an annual rate of 6% compounded continuously. How long does it take: • To double your money? *Divide both sides by P *ln both sides *Apply Law 4 *lne = 1 *Solve for t

Additional Example An investment of $500 is made at 3. 6% annual interest. How long does it take to triple the investment: A) if it is compounded monthly? B) if it is compounded continuously?

The Change of Base Formula The change of base formula can be used to find the value of the exponent if it is unknown. Determine the value of x: 2 x=30 Rewrite using the definition of log: Use the change of base formula to change to base 10:

The Change of Base Formula

HW: • Section 5 -6, Pg. 200, 3 -33 multiples of 3 • and • Section 5 -7, P 205: 9 -19 odd