7.2 Logarithmic Functions Name: 1

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1 Write your questions and thoughts here 7.2 Logarithmic Functions Name: 1 Definition of Logarithm: y = log x if and only if b = x b > 0, b 1, x > 0 How does Mr. Bean say log x? How does Mr. Kelly say log x in his head? Recall: An function is of the form y = b A function is of the form x = b These functions are inverse functions Rewrite in exponential form: Rewrite in logarithmic form: 1. log 125 = 3 2. = Evaluate the following logarithms (without a calculator): 3. log 8 4. log " 5. log 1 PROPERTIES OF LOGARITHMS Product Property: Quotient Property: Power Property: log xy = log x + log y log = log x log y log x = ylog x Expand: EXPANDING AND CONDENSING LOGRITHMS: Condense: 5 6. log 4 3x 5 p y 7. log 3 q log 4 8 log log 4 25 CHANGE- OF- BASE: Solve for x: Bean: (take log both sides) Method: Brust: (cancel by using log of base) Sully: JUST GRAPH IT, BABY log C a = log b a log b c = 7 9. Evaluate log " Evaluate log = 2

2 Write your questions and thoughts here 7.2 Logarithmic Functions Other Reminders: log b = log 1 = log " x is called: log b = b "# = log x is called: Solve for the unknown variable. 14. log x = GRAPHING LOGARITHMIC FUNCTIONS: 15. ln x = Use the properties of logarithms to find the inverse of the given function. (Hint Switch x and y and solve for y) 16. f(x) = f(x) = f(x) = f x = log x ln y = 2 ln x Re

3 7.2 Practice LOGARITHMIC FUNCTIONS Name: Pre- Calculus You might as well get these bad boys out of the way first. Solve for each unknown variable Quick Review For 4-6, Expand the logarithm. (NOT L I K E T H I S ) 4. log ab c 5. ln 6. log " For 4-6, Rewrite the expression as a single log. (C o n d e n se) 7. log a 2 log b + 3 log c 8. 2 ln x + 5 ln y ln z 3 9. log y + 7log x + "# Solve for x using the Bean method (change of base formula). Show your work Go out four places = = = 1000 Solve for x by using the Brust method (canceling the base with logs). Show your work Go out four places = = = 50 Solve for x by using the Sully method (by graphing). Tell the point of intersection used to solve the equation = = = 10 x = Point (, ) x = Point (, ) x = Point (, )

4 Find x, y, or b as indicated in the following problems. 19. log x = log " 8 = y 21. log 16 = log 1 = log x = 24. log 9 = y 25. log 1000 = Use logarithms to find the inverse of the given function. 26. f x = f x = f x = f x = ln (3x) 30. log y = 3 log x *. log y = "# 32. Condense into a single logarithm. "# log y Expand. * ln x3 y z

5 7.2 Application and Extension 1. Expand: log 2. Solve for x: 3 x = When Sully is ready to retire, he has plans of moving to New York City to become a butcher. In fact, he wants to open his own butcher shop, The New York Metzgerei, where he can sell his signature product: Sullamy Picante Sully has to cook the meat and then let it cool while recording the temperature during the production process. One day, Sully observes the following temperatures: Time (min) Temperature (degrees above room temp in F) a. Plot the data on the graph to the right. Enter Time into L 1 and Temp into L 2.. b. Would a linear model be appropriate for this data? Why or why not? c. To straighten the data, take the common log of each of the temperatures. (Log L 2 à L 3 ) Complete the table: Diff from Room Temp ( o F) 10 o 0 o 10 Time (Min) Time (min) Log (Temp) d. Calculate the linear regression for Time vs. Log Diff Temp. Plot the data and graph line of best fit on the graph. (For help with your calculator, watch the Application Help Video) e. Is the data straighter? Log Temp ( o F) 1.0 (Flip it like an Algebro)

6 f. Now complete a LINEAR REGRESSION using your calculator on Time and Log Temp. Write your linear equation below, accurate to 4 decimal places. Remember, we aren t using y, we are using log y. (Use LinReg L 1, L 3 ) a = Log y = ax + b b = Log y = g. In statistics, straighter data leads to more accurate predictions. We take the log of the dependent variable to straighten out exponential data. But you know what? We like the ORIGINAL variable and we hate equations with logs in them. Let s use our Log rules to reverse transform the equation into an exponential equation. For even more fun, let s bust out the 2 column proof: Regression Equation Statements Reasons 1. Log y = 1. Given Your equation from above Raise 10 to the power of each side of the equation Write the exponent sum as the product of a common base. y- intercept value Compute Rewrite the exponent product as a power to a power Compute 10 slope of equation Rewrite your equation in the form y = ab x h. Confirm your exponential regression equation with ExpReg in your calculator. (Statà Calcà ExpReg L 1, L 2.) Be sure you use the original data and not the transform data in L 3. Congrats You just learned how the calculator calculates an exponential regression call it The Great Regression of 7.2 i. How hot was the Sullamy Picante when Sully took it out of the oven? (Assume a room temperature of 72 o ) Use your equation to figure it out. Show your work below.