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5-Minute Check on Activity 5-9

Learn about logarithms and exponential regression functions, including how to determine regression equations, predict values, and apply logarithmic and exponential forms. Explore the properties of logarithms and practice solving related problems.

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5-Minute Check on Activity 5-9

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  1. 5-Minute Check on Activity 5-9 Using the exponential regression function in your calculator to two-decimal places of accuracy: Determine the regression equation for the following data: Use that regression equation to predict what y(6) = Determine the regression equation for the following data: Use that regression equation to predict what y(6) = y =2(2.55)x y(6) = 549.88 y =3.48(0.60)x y(6) = 0.16 Click the mouse button or press the Space Bar to display the answers.

  2. Activity 5 - 10 The Diameter of Spheres

  3. Objectives • Define logarithm • Write an exponential statement in logarithmic form • Write a logarithmic statement in exponential form • Determine log and ln values using a calculator

  4. Vocabulary • Logarithms – are inverses of exponential functions. The equation logb x = y is equivalent to by = x. • Common logarithms – are logarithms to the base 10 • Natural logarithms –are logarithms to the base e • e – is the natural number, apx 2.712818

  5. Activity Spheres are all around us (pardon the pun). You play sports with spheres like baseballs, basketballs, and golf balls. You live on a sphere, the Earth; which is orbited by a sphere, the moon; and both of them orbit another sphere, the sun. All spheres have properties in common. For example, from Geometry, the formula for volume is V = 4/3πr3 and the formula for surface area is SA = 4πr2 (where r represents the radius of the sphere). But not all spheres are the same size!

  6. Activity cont The table below has the diameter (remember d = 2r) of several spheres mention before: If we wanted to plot diameter as the x-variable and either volume or surface area as the y-variable, then we would need to put all these values on the horizontal axis some how.

  7. Activity cont If we plot the first three on the axis below If we plot the last three on the axis below 0 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000

  8. Logarithmic Scale If we scale the axis in exponential powers of 10 (each tick mark is ten times larger than the one before), then we can fit them all on the same scale. This is called a logarithmic scale. Many times a logarithmic scale will be marked with just the exponents (and a note about it being logarithmic). 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108 109 1010 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 (logarithmic scale)

  9. Richter Scale - Earthquakes The magnitude value is proportional to the logarithm of the amplitude of the strongest wave during an earthquake. A recording of 7, for example, indicates a disturbance with ground motion 10 times as large as a recording of 6. The energy released by an earthquake increases by a factor of 30 for every unit increase in the Richter scale. The table below gives the frequency of earthquakes and the effects of the earthquakes based on this scale

  10. Richter Scale - Earthquakes

  11. Some Properties of Logarithms In general, a statement in logarithmic form is logb x = y, where b is the base of the logarithm, x is a power of b, and y is the exponent. The base b for a logarithm can be any positive number except 1. Common logs are to the base 10 and natural logs are to the base e. The equation logb x = y is equivalent to by = x. If b > 0 and b ≠ 1 then Logb bn = n Logb 1 = 0 Logb b = 1

  12. Properties of Logarithms Examples = 4 log 10 = 4(1) = 4 Evaluate each of the following: • Log 104 • Log (1/100) • Log 1000 • Log4 64 • Ln e • Log3 (1/27) = log 10-2 = -2 log 10 = -2(1) = -2 = log 103 = 3log 10 = 3(1) = 3 = log4 43 = 3log4 4= 3(1) = 3 = loge e = 1 = log3 3-3 = -3log3 3= -3(1) = -3

  13. Properties of Logarithms Examples = 1.30103 Evaluate each of the following with your calculator: • Log 20 • Ln 15 • Log 0.02 Rewrite each logarithmic expression as exponential: • 3 = log2 8 • 0 = log5 1 = 2.7081 = -1.699 23 = 8 50 = 1

  14. Properties of Logarithms • Logs of Products can be simplified to the sum of the logs • Logs of Quotients can be simplified to the difference of the logs • Logs of Exponents can be simplified to the product of the exponent and the log log ab = log a + log b log a/b = log a – log b log ab = b log a

  15. Properties of Logarithms Examples = 4 log x Evaluate each of the following: • Log x4 • Log (1/y) • Log 10x • Log x/y • Ln x6y4 • Ln (x2z7 / y5) = log y-1 = - log y = log 10 + log x = 1 + log x = log x – log y = 6 ln x + 4 ln y = 2 ln x + 7 ln z – 5 ln y

  16. Summary and Homework • Summary • Notation of logarithms is y = logb xwhere b is the base of the log, x is the resulting power of b and y is the exponent • Common logs (base 10) y = log x • Natural logs (base e) y = ln x • y = logb x is equivalent to exponential by = x • Special log properties: • logb 1 = 0 • logb b = 1 • logb bn = n • Homework • pg 632 – 633; problems 1- 5

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