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Properties of Logarithms

Properties of Logarithms

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Properties of Logarithms

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  1. Properties of Logarithms If we want to evaluate a logarithm on the calculator, we may need to change the base Ex. Evaluate log425

  2. Properties of Logarithms These apply to all logarithms, not just the common log

  3. Ex. Rewrite in terms of ln 2 and ln 3 a) ln 6 b)

  4. Ex. Find the exact value without a calculator a) b)

  5. Ex. Expand each logarithmic expression a) b)

  6. Ex. Condense each logarithmic expression a) b)

  7. Practice Problems Section 5.3 Problems 9, 17, 23, 47, 69

  8. Exponential and Logarithmic Equations Ex. Solve for x a) 2x = 32 b) ln x – ln 3 = 0 c)

  9. Ex. Solve for x d) ln x = -3 e) log x = -1 f)

  10. Ex. Solve for x g) h) ex + 5 = 60

  11. Ex. Solve for x i) 5e2x = ex – 3

  12. Ex. Solve for x j)

  13. Ex. Solve for x k)

  14. Ex. Solve for x l) ln x = 2 m) log3(5x – 1) = log3(x + 7) n) log6(3x + 14) – log65 = log62x

  15. Ex. Solve for x p) 5 + 2ln x = 4 q) 2log53x = 4

  16. Ex. You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?

  17. Practice Problems Section 5.4 Problems 13, 25, 51, 53, 67, 77, 85, 87, 99

  18. Exponential and Logarithmic Models Exponential Growth Exponential Decay

  19. Ex. The number of households, D, in millions, with HDTV is given by the exponential model D = 30.92e0.1171t, where t is years since 2000. When will the number of households reach 100 million?

  20. Ex. The number N of bacteria in a culture is modeled by N = 450ekt, where t is time in hours. If N = 600 when t = 3, estimate the time required for the population to double.

  21. Ex. A population of fruit flies is increasing according to an exponential growth model. After 2 days, there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days?

  22. Ex. A picture supposedly painted in approximately 1675 contains 99.5% of its carbon-14 (half-life 5730 years). Is the painting a fake?

  23. Gaussian Model This is often used in probability and statistics to represent a normally distributed (bell-shaped) model

  24. Logistic Growth Model This can be used to represent a model with a rapid increase followed by leveling off.

  25. Ex. On a college campus of 5000 students, the spread of a virus is modeled by where y is the number of students infected after t days. • How many students are infected after 5 days? b) If the college cancels classes when 40% of students are infected, after how many days will classes be cancelled?

  26. Practice Problems Section 5.5 Problems 35, 37, 47, 49