Common Logs and Applications Unit 3 Day 13 Common Logs and Applications

Warm Up y = log10 (4x – 11) – 2 x > 2.75 (11/4) All Real #s Key Features of Logarithmic Functions Graph the following function. Identify the domain, range, asymptotes, and transformations of the parent function y = log (x). y = log10 (4x – 11) – 2 Domain: Range: Asymptote: Description of transformations: x > 2.75 (11/4) All Real #s x = 2.75 Right 2.75 and down 2 Continue to chart on p. 33  a. 100 b. 10,000 c. 1,000,000 d. 0.01 e. -0.001 f. 3.45 g. -34.5 h. 345 i. 0.0023

Warm Up Express each of the numbers below as accurately as possible as a power of 10. a. 100 10x=100, x=2 b. 10,000 c. 1,000,000 d. 0.01 e. -0.001 f. 3.45 g. -34.5 h. 345 i. 0.0023 10 2 10 4 10 6 10 -2 -10 -3 10 0.538 -10 -1.538 10 2.538 10 -2.63

Homework Answers – Day 12 1. a) Translated 2 units right. b) Translated 3 units up. c) Translated 2 units right and 3 units up. 2. Domain : x > 2. Range : All real numbers. x-intercept (2, 0) Vertical Asymptote: x = 2 3. a) Reflected over the x-axis. b) Translated 2 units right. c) Reflected over the x-axis and translated 2 units right. Domain : x > 2. Range : All real numbers. x-intercept (3, 0) Vertical Asymptote x = 2

Homework Answers – Day 12 5. a) Reflected over the y-axis. b) Translated 2 units up. c) Reflected over the y-axis and translated 2 units up. Domain : x < 0. Range : All real numbers. x-intercept: (-0.01, 0) Vert. Asymptote: x = 0 7. #7 Graph Scale: x-values increase by 10, y values increase by 1.

Homework Answers – Day 12 8. #8 Graph Scale: x values increase by 10, y values increase by 1.

Investigation 1 Notes p. 33 & 34 Unit 3 Day 13 Investigation 1 Notes p. 33 & 34

Common Logarithms We have already discussed logarithms in the previous lesson, but in this lesson we will solve exponent problems which focus on a base of 10. When we solve logarithmic problems in base 10, we call them common logarithms. The definition of common logarithms is usually expressed as: log10 a = b if and only if 10b = a Log10 a is pronounced “log base 10 of a”.

You Try. 1. Use your calculator to find the following logarithms You Try! 1. Use your calculator to find the following logarithms. Then compare the results with your work on Problem 1. = 2 because 102 = 100 = 4 104 = 10,000 a. log 100 b. log 10,000 c. log 1,000,000 d. log 0.01 e. log -0.001 f. log 3.45 g. log -34.5 h. log 345 i. log 0.0023 = 6 106 = 100,000,000 = -2 10-2 = 0.01 = 0.538 100.538 = 3.45 Non real answer! Because domain x > 0 = 2.538 102.538 = 345 = -2.638 10-2.638 = 0.0023 Non real answer! Because domain x > 0 Continue to questions # 2  >

2) What do your results from Problem 2 (especially Parts e and g) suggest about the kinds of numbers that have logarithms? See if you can explain your answer by using the connection between logarithms and the exponential function y = 10x. Domain of logs must be positive because a base that is positive raised to a power can never be a negative number.

Summarize the Mathematics A) How would you explain to someone who did not know about logarithms what the expression log y = x tells us about the numbers x and y? B) What can be said about the value of log y in each case below? Give brief justifications of your answers. i. 0 < y < 1 _________________________________________ ii. 1 < y < 10 ________________________________________ iii. 10 < y < 100 ______________________________________ iv. 100 < y < 1,000_____________________________________

Use your understanding of the relationship between logarithms and exponents to help complete these tasks. Find these common (base 10) logarithms without using a calculator. i. log 1,000 ii. log 0.001 iii. log 103.2 Use the function y = 10x, but not the logarithm key of your calculator to estimate each of these logarithms to the nearest tenth. Explain how you arrived at your answers. i. log 75 ii. log 750 iii. log 7.5 3.2 3 -3 1.88 2.88 0.88

Common Logs: Solving for Exponents and Word Problems Unit 3 Day 13 Part 2 Common Logs: Solving for Exponents and Word Problems

Use number sense and what you already know about logarithms to solve these equations. 10x = 1,000 b. 10x + 2 = 1,000 c. 103x + 2 = 1,000 d. 2(10)x = 200 x = 1 x = 3 x = 1/3 x = 2

= You Try! e. 3(10)x + 4 = 3,000 f. 102x = 50 g. 103x + 2 = 43 h. 12(10)3x + 2 = 120 i. 3(10)x + 4 + 7 = 28 x = -1 x = 0.85 x = -.12 x = -1/3 x = -3.15

Complete Investigation p. 35-36 Show under document cam or circulate room to check student responses to application problems.

Practice p. 36

b. The population of the United States in 2006 was about 300 million and growing exponentially at a rate of about 0.7% per year. If that growth rate continues, the population of the country in year 2006 + t will be given by the function P(t) = 300(100.003t). According to that population model, when is the U.S. population predicted to reach 400 million? Check the reasonableness of your answer with a table or a graph of P(t). t = 41.65 2006 + 41.65 = 2047.65 Year 2047

Homework p. 19 ODD and p. 20 EVEN

Review Review (not in notes) Find the solution to each equation algebraically. 1) 2) 3) 4) Your new painting is valued at $2400. It’s value depreciates 7% each year. The value is a function of time. Write a recursive (next-now) equation for the situation Write an explicit function for the situation When will the painting be worth $1000? 5) Graph y =2x+4 - 3. Identify the domain, range, asymptotes, and transformations of the parent function y =2x . Review if there is extra time or save for Day 14!

Review ANSWERS x = 3 x = 1002 x = -3 Review (not in notes) Find the solution to each equation algebraically. 1) 2) 3) x = 3 x = 1002 x = -3

Review ANSWERS NEXT = Now * (.93) start = 2400 y = 2400(.93)x 4) Your new painting is valued at $2400. It’s value depreciates 7% each year. The value is a function of time. Write a recursive (next-now) equation for the situation Write an explicit function for the situation When will the painting be worth $1000? 5) Graph y =2x+4 - 3. Identify the domain, range, asymptotes, and transformations of the parent function y =2x . NEXT = Now * (.93) start = 2400 y = 2400(.93)x During year 12 Domain: All real #s HA: y = -3 Range: y > -3 Transformation: Left 4, down 3