1. 1. Use a property of logarithms to evaluate log 3 9 2 2. Use log 5 ≈ 0.699 and log 6 ≈ 0.778 to approximate the value of log 150. 3. Expand ln 7 3 2x 4. 4. Condense 2log 5 y – 2log 5 5 + ½log 5 x. 5. Use the change of base formula to evaluate log 8 20. 1 Algebra II

2. Solving Exponential and Logarithmic Equations Algebra II

3.  To solve equations you must undo what is being done  Exponential and logarithmic functions are inverse operations  You undo “exponentiating” by ”logarizing” and you undo “logarizing” by “exponentiating” 3 Algebra II

4.  a x = a y If x = y  Log b x = log b y if x = y Examples: If 2 x = 2 3 then x = 3 If ln x = 3 then e lnx = e 3, so x = e 3 If log 3 x = log 3 (5), then x = 5 If e x = 7 then ln e x = ln7, so x = ln7 4 Algebra II

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12. 12 Algebra II No Solution (Extraneous solution)

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17. 17 Algebra II Check both solutions!

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23.  ln x = x 2 – 2  Log 3 x = x 2 – 2 23 Algebra II

24. The population size y of a community of lemmings varies according to the relationship y = y 0 e 0.15t. In this formula, t is time in months and y 0 is the initial population at the time 0. Estimate the population after 8 months if there were originally 2000 lemmings. y = y 0 e 0.15t y = 2000e 0.15(8) y = 2000e 1.2 y ≈ 6640.2339 In 8 months, the population will be approximately 6640 lemmings. Algebra II 24

25. How long does it take an investment of $5,000 to double if it is invested at 4%, compounded quarterly? P = $5000 r = 4% or 0.04 Compounded quarterly = 4 times per year, n = 4. The investment doubles, so A must be $10,000. Substitute these values and solve for t. Algebra II 25

26. It takes more than 17 years for the money to double in value. Algebra II 26

27. 27 Algebra II Solve each equation. Be sure to check solutions!! 1.3 2x = 27 x+2 2.5e 3x + 2 = 17 3.log 4 (5x – 11) = log 4 (3 – 2x) 4.-3 ln = 4