Chapter 10.4 Common Logarithms Standard & Honors Algebra II Mr. Gilbert Chapter 10.4 Common Logarithms Standard & Honors 9/19/2018

Agenda Warm up Work book (participation grade) Lesson Homework 9/19/2018

Homework Review 9/19/2018

Communicate Effectively pH level: hydrogen ion h+ concentration 1.0 battery acid, - 4.2 Tomatoes, - 5.0 Black Coffee 7.0 Pure Water 7.8 Eggs, - 10.0 Milk of Magnesia, 13.5 Lye pH Formula Erg: measure of energy and mechanical work, the force required to accelerate a mass of one gram at a rate of one centimeter per second squared. Richter Magnitude Formula M = 2/3 log10 E | E seismic energy in 10,000,000 ergs. 9/19/2018

Examples: Richter Magnitude Scale (do not copy) -1.5 Breaking a rock on a lab table 1.0 Large Blast at a Construction Site 2.0 Large Quarry or Mine Blast 4.0 Small Nuclear Weapon 4.5 Average Tornado (total energy) 6.0 Double Spring Flat, NV Quake, 1994 7.0 Largest Thermonuclear Weapon 8.0 San Francisco, CA Quake, 1906 9.0 Chilean Quake, 1960 10.0 (San-Andreas type fault circling Earth) 12.0 (Fault Earth in half through center) Source: http://staff.imsa.edu/science/geophysics/geosphere/tectonics/eqenergy.html 9/19/2018

Example 1 Find Common Logarithms (2) Example 2 Solve Logarithmic Equations Using Exponentiation (3) Example 3 Solve Exponential Equations Using Logarithms (2) Example 4 Solve Exponential Inequalities Using Logarithms (3) Example 5 Change of Base Formula (2) Formula 9/19/2018

Use a calculator to evaluate log 6 to four decimal places. ENTER LOG Keystrokes: 6 .7781512503 Answer: about 0.7782 Use a calculator to evaluate log 0.35 to four decimal places. ENTER LOG Keystrokes: 0.35 –.4559319557 Answer: about –0.4559 9/19/2018 Example 4-1a

Use a calculator to evaluate each expression to four decimal places. a. log 5 b. log 0.62 Answer: 0.6990 Answer: –0.2076 9/19/2018 Example 4-1c

Write each side using 10 as a base. Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log The San Fernando Valley earthquake of 1994 measured 6.6 on the Richter scale. How much energy did this earthquake release? Write the formula. Replace M with 6.6. Simplify. Write each side using 10 as a base. 9/19/2018 Example 4-2a

Inverse Property of Exponents and Logarithms Use a calculator. Answer: The amount of energy released was about ergs. 9/19/2018 Example 4-2b

Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log In 1999 an earthquake in Turkey measured 7.4 on the Richter scale. How much energy did this earthquake release? Answer: about 9/19/2018 Example 4-2c

Property of Equality for Logarithmic Functions Solve Original equation Property of Equality for Logarithmic Functions Power Property of Logarithms Divide each side by log 62. Use a calculator. Answer: 9/19/2018 Example 4-3a

Check You can check this answer by using a calculator or by using estimation. Since and the value of x is between 2 and 3. Thus, 2.5643 is a reasonable solution. Solve Answer: 2.5789 9/19/2018 Example 4-3c

Solve Original inequality Property of Inequality for Logarithmic Functions Power Property of Logarithms Distributive Property Subtract 5x log 3 from each side. 9/19/2018 Example 4-4a

Use a calculator. Simplify. Distributive Property Divide each side by Switch > to < because is negative. Use a calculator. Simplify. 9/19/2018 Example 4-4b

Negative Exponent Property Check: Original inequality Replace x with 0. Simplify. Negative Exponent Property Answer: The solution set is 9/19/2018 Example 4-4d

Solve Answer: 9/19/2018 Example 4-4e

Answer: The value of is approximately 2.6309. Express in terms of common logarithms. Then approximate its value to four decimal places. Change of Base Formula Use a calculator. Answer: The value of is approximately 2.6309. 9/19/2018 Example 4-5a

Express. in terms of common logarithms Express in terms of common logarithms. Then approximate its value to four decimal places. Answer: 9/19/2018 Example 4-5b

Homework See Syllabus 10.4 9/19/2018