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Chapter 10: Exponential and Logarithmic Relations

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Chapter 10: Exponential and Logarithmic Relations

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    1. Chapter 10: Exponential and Logarithmic Relations Continuing focus on functions (domain, range, graphs) Lots o’ word problems ? The new concept of logarithms: allow us to solve problems like 3x = 11

    2. 10. 1 Exponential Functions Graphing an exponential function Definition and features Growth vs. decay Word problems Working with irrational exponents Solving for variable in an exponent position (déjà vu)

    3. Example 1-1a

    4. Example 1-1b

    5. comments Notice that in the previous graph: as x ? -8, y ? 0 That is, as x gets smaller, the y value gets closer and closer to the x-axis.. BUT it will never actually hit the x-axis, it just keeps getting closer We say that the x-axis is an asymptote – a line that the graph of a function/relation gets closer and closer to but never touches

    6. Example 1-1c

    7. More about exponential functions Any function of the form y = a* bx, where: a ? 0 and b > 0 but b ? 1 is considered an exponential function with a base of “b” Why couldn’t b = 1? Because 1 raised to any power is still 1, so that if b = 1 you have a constant function

    8. Other features of exponential functions They are continuous (no gaps/jumps) and one-to-one (pass vertical AND horizontal line tests) The domain is all real numbers The x-axis is an asymptote for the graph (this can change if we add a constant to the function, causing a vertical translation) The range is the set of all positive numbers if a > 0 and is the set of all negative numbers if a < 0 The y-intercept is at (0,a) The graphs of y = abx and y = a (1/b)x are reflections across the y-axis

    9. Growth vs. decay If b > 1, then the y-values increase as x gets larger and we say the function represents exponential growth If 0 < b < 1, (for example, b = ½ ) then the y-values decrease as x gets larger and we say the function represents exponential decay

    10. Example 1-2a

    11. Example 1-2b

    12. Example 1-2c

    13. Example 1-2d

    14. Word problems If you have a problem where a quantity is increasing or decreasing by a certain percentage in a given time period, use the following model for the word problem: Y = a (1 ± r)t, where a is the starting amount, r is the rate (in decimal form) that the quantity increases or decreases by, and t is the # of time periods (it is basically the x-variable in these problems For example, if the population of a town is currently 32,000 and it is projected to increase by 5% each year, we could use the formula: Y = 32000 (1 + .05)t or y = 32000 (1.05)t to predict the population after t years

    15. Example 1-3a

    16. Example 1-3b

    17. Example 1-3c

    18. Example 1-3d

    19. Example 1-3e

    20. Working with Irrational Exponents Just use the regular laws for exponents! Ax * Ay = Ax + y Ax ? Ay = Ax – y (Ax)y = Axy

    21. Example 1-4a

    22. Example 1-4b

    23. Example 1-4c

    24. Solving equations where variable is in the exponent position We’ve done these before! Just rewrite each side using the same base For example, if 32 = 2x + 9, 1st rewrite as: 25 = 2x + 9 Then set the exponent expressions equal to one another and solve!

    25. Example 1-5a

    26. Example 1-5b

    27. Example 1-5c

    28. Example 1-5d

    29. Example 1-6a

    30. Example 1-6b

    31. Example 1-6c

    32. HOMEWORK!!! Pg. 528 – 529 #s 24 – 44 evens 48 – 56 evens 59 – 60, 62 – 64

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