Algebra 2 Module 3 Invest. Test Review

Module 3 Invest. 6-8 Test Review Algebra 2 Review Review Review Review Review **SHOW ALL WORK!** Rewrite the following in exponential form. 1. log₁₅(1/225) = -2 15⁻² = 1/225 2. log₂(x) = y 2ʸ = x Rewrite the following in logarithmic form. 3. 6³ = 216 log₆(216) = 3 4. 12ˣ = y log₁₂(y) = x Condense or rewrite the following as a single logarithm. Simplify, if possible. 5. log₆c + log₆c + log₆c + log₆c = log₆(c·c·c·c) = log₆(c⁴) = 4log₆(c) 6. log₁₀10 + log₁₀5 + ½log₁₀25 = log₁₀(10) + log₁₀(125) + log₁₀(√25) = log₁₀(10·125·5) = log₁₀(6250) 7. ½log₉9 - log₄4 = log₉9^(1/2) - log₄4 = log₉3 - log₄4 = log(3/4) 8. log₅250 - log₅2 = log₅(250/2) = log₅(125) = log₅5³ = 3log₅5 = 3 9. ln(200) - ln(10) - ln(5) = ln(200/10) - ln(5) = ln(20) - ln(5) = ln(20/5) = ln(4) Expand the following. 11. log₃(x²/(ab)) = log₃(x²) - log₃(a) - log₃(b) = 2log₃(x) - log₃(a) - log₃(b) 12. ln(2x⁵) = ln(2) + ln(x⁵) = ln(2) + 5ln(x) 13. log₂(7c/d) = log₂(7c) - log₂(d) = log₂(7) + log₂(c) - log₂(d) Evaluate each of the logarithms. 14. log₂(1/8) = -3 15. log₆(36) = 2 16. log₈(8) = 3 17. log₇(1/49) = -2 18. log(1000) = 3 Complete each table of values and graph on the next page. 19. | x | f(x) = 8^x | |---|---| | -1 | 1/8 | | 0 | 1 | | 2/3 | 4 | | 1 | 8 | | 2 | 64 | 20. | x | g(x) = log₂(x) | |---|---| | 1/4 | -2 | | 1/2 | -1 | | 1 | 0 | | 2 | 1 | | 4 | 2 | [Image of two graphs] 21. Use the properties of logarithms to rewrite 1/3 log₄8 + 7 log₄2 as a single logarithm. Then evaluate the value of the resulting expression. = log₄8^(1/3) + 7 log₄2 = log₄2 + log₄2^7 = log₄2 + log₄128 = log₄(2 * 128) = log₄(256) = 4 Module 3 Invest. 6-8 Test Review Algebra 2 Review Review Review Review Review **SHOW ALL WORK!** Solve the following equations. Use logarithms and show all work algebraically! Round your final answer to the thousandths.   22. 8ⁿ = 500 log₈(500) = n log(500)/log(8) = n 2.989 = n 23. 3ˣ/2 = 17 3ˣ = 8.5 log₃(8.5) = x log(8.5)/log(3) = x 1.948 = x 24. 1000/781 = 781(1.056)^t 1.28 = 1.056^t log(1.28) = t * log(1.056) t = log(1.28)/log(1.056) t = 4.521 Evaluate each of the following logarithms. Use your calculator and round your final answer to the thousandths. 25. log₄(67) = 3.033 26. log₇(123) = 2.493 27. log(400) = 2.602 28. ln(14) = 2.639 **Problem 3:** The population of Chandler grew from 176,585 people in 2004 to 211,902 people in 2014. a) Define an exponential function that models the town's population as a function of the number of years since 2004. Be sure to define your variables. f(x) = 176585(1.018)^x x: # of years since 2004 b) What is the annual growth factor and the annual percent change? Round your final answers to the thousandths. annual growth factor = 1.018 1.018 = 1 + percent change 0.018 = percent change 1.8% percent change c) Use your function to predict the city's population in 2025. Round your final answer to the nearest whole   number. 2025 - 2004 = 21 x = 21 f(21) = 176585(1.018)^21 = 256,837 d) According to your function, when will the population be 290,000? Use logarithms and solve algebraically. Round your final answer to the thousandths.   290000 = 176585(1.018)^x 1.642 = 1.018^x log(1.642) = x * log(1.018) x = log(1.642)/log(1.018) x = 27.198