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Algebra II Unit 9 Name: __________________ Date: _________ Per: _____ Directions: Follow the directions for each task. Anything not completed is homework (due tomorrow). Tomorrow we will begin class with an entry slip. Task #1: Review your notes from Monday / Copy into notes if you don’t have them II. Logarithms vs. Exponents a. Vocabulary The first two (i and i. Logarithm: If y = bx, then logby = x. Remember “base raised to the ii) are converting b. Examples answer equals the inside.” from exponential i. 25 = 52 => log525 = 2 form to logarithmic 3 ii. 1/8 = (1/2) => log1/21/8=3 form. The next two iii. log41 = 0 => 40 = 1 (iii and iv) are converting from iv. y = log (x – 5) => 10y = x – 5 logarithmic form to exponential form. 10 Task #2: Write the opposite form in the space provided below. 1. log232 = 5 ______________ 2. 32 = 9 ______________ 3. log51 = 0 ______________ 4. x5 = (y – 2) ______________ 5. log 10 = 1 ______________ 6. log1/2 2 = -1 ______________ 7. (1/4)-1 = 4 ______________ 8. log 0.1 = -1 ______________ 9. 6-2 = 1/36 ______________ 10. 100 = 1 ______________ Task #3: Review your notes from Monday / Copy into notes if you don’t have them c. Examples (Evaluating logarithms) i. log816 log816 = x Steps: 8x = 16 1. Write an equation in logarithmic (23)x = 24 form 23x = 24 2. Convert to exponential form 3x = 4 3. Write each side using the same base x = 4/3 4. Power Property of exponents ii. log927 log927 = x 9x = 27 (32)x = 33 32x = 33 2x = 3 x = 3/2 (distribute) 5. Set the exponents equal to each other 6. Solve for x. iii. log64 1/32 log64 1/32 = x 64x = 1/32 (26)x = 2-5 26x = 2-5 6x = -5 x = -5/6 Think… 1 1 5 25 32 2 Task #4: Evaluate each logarithm. Show all work below. 1. log48 2. log497 3. log5 (1/25) 4. log 0.01 = -2 5. log8127 6. log1/4 (1/8) Task #5: NEW NOTES! Copy this into your notes. III. Expanding and Condensing Logarithms a. Properties of logarithms For any positive numbers, M, N, and b, b 1: logbMN = logbM + logbN (Product Property) M logb = logbM – logbN (Quotient Property) N logbMx = xlogbM (Power Property) b. Examples of condensing i. log320 – log34 20 log3 4 log35 Use the quotient property. Then simplify. ii. 3log2x + log2y log2x3 + log2y log2 (x3y) iii. 3log 2 + log 4 – log 16 log 23 + log 4 – log 16 log 8 + log 4 – log 16 8 4 log 16 32 log 16 log 2 Use the both the power and product properties. Always work left to right. Remember to simplify as much as possible. THINK BOX Can you write 3log2 9– log6 9 as a single logarithm? Why or why not? Task #6: Independent practice. Show all work below. 1. log24 + 3log29 2. log7a + log7b – 3log7c 3. 3log a – 2 log b 4. ½ log x – log y 5. 2 log3 3 + log35 6. log4 64 – log4 16 Task #7: NEW NOTES! Add this to your notes. c. Examples of expanding x i. log5 Use the quotient property. y log5 x - log5 y ii. log 3r4 log 3 + log r4 log 3 + 4log r y 2 iii. ln 3 y2 ln 2 3 2 ln y – ln 32 2 ln y – 2 ln 3 Use the product and power properties. Simplify by distributing the exponent. Then use quotient and power properties. Is there another way to do this one? Task #8: Independent practice. Show all work below. a 1. log3 2. log (3x3y2) b 3. log (2x)-2 4. log5 x 3 y 2 z 5. ln ab c 2 6. log27b Any questions? Write them here. More practice problems on the website: sternmass.org