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7-4A Introduction to Logarithms Name:__________________ 1. Guess the exponents. 10 x 1000 5b 625 3z 81 x b z 1 t ) 9 81 t 125a 5 7 d 343 16 w 64 a d w 10 y 1 100 y ( 2. Find a pattern. Exponential Form Logarithmic Form 25 32 34 81 42 1 16 53 125 102 1 100 811/2 9 3. Did you notice any pattern? Can you create a rule? base(exponent) result log (base ) (result ) exponent 4. Definition of a Logarithm: The logarithm of a number is the exponent by which another fixed number (= the base) must be raised to produce that number. In general if b n x, then logb x n. 5. Practice problems 1) log 2 16 x 2) log12 y 2 Alg 2 Adv Ch. 7B Notes-page 1 3) logb 216 3 4) log 2 1 a 32 5) log3 3 z 7-4B Evaluate Logarithms and Graph Logarithmic Functions Objective: To evaluate logarithms and graph logarithmic functions Algebra 2 Standards 11.1 and 14.0 *Definition of Logarithm with Base b: Let b and y be positive numbers with b ≠ 1. The logarithm of y with base b is denoted by logb y and is defined as follows: logb y x if and only if bx y The expression logb y is read as “ log base b of y.” The equations logb y x and b x y are equivalent. The first is in logarithmic form and the second is in exponential form. Ex. 1: Rewrite logarithmic equations in exponential form. a. log 2 32 5 b. log10 1 0 c. log9 9 1 d. log1 5 25 2 You Try: Rewrite the equation in exponential form. a. log3 81 4 b. log7 7 1 c. log14 1 0 d. log1 2 32 5 Ex. 2: Evaluate the logarithms. a. log3 81 b. log10 0.001 c. log64 2 You Try: Evaluate the logarithms. a. log 2 32 b. log 27 3 c. log1 4 256 *Special Logarithms: A common logarithm is a log with base 10. It is denoted by log10 or simply by log. A natural logarithm is a log with base e. It is denoted by log e or more commonly by ln. Common Logarithm log10 x log x Natural Logarithm loge x ln x Ex. 3: Evaluate common and natural logs. Use a calculator to evaluate the logarithm. (Round to the nearest thousandths) a. log 0.85 b. ln 22 You Try: Use a calculator to evaluate the logarithm. (Round to the nearest thousandths) a. log 12 b. ln 0.75 Alg 2 Adv Ch. 7B Notes-page 2 Ex. 4: The sales of a certain video game can be modeled by y 20ln x 1 35 , where y is the monthly number (in thousands) of games sold during the xth month after the game is released for sale (x > 1). Estimate the number of video games sold during the 10th month after the game is released. *Inverse Functions: By definition of a logarithm, it follows that the logarithmic function g x logb x is the inverse of the exponential function f x b x . This means that: g f x logb b x x and f g x blogb x x Ex. 5 Using Inverse properties. Simplify the expression. a. eln 9 b. log 3 27 x b. log 2 64 x c. 8log8 x c. e ln 20 You Try: Simplify the expression. a. log 7 7 3 x Ex. 6 Find Inverse functions. a. y 8 x b. y ln x 4 You Try: Find the inverse. a. y 4 x b. y ln x 5 *Parent graphs for Logarithmic Functions Because f x logb x and g x b x are inverse functions, the graph of f x logb x is the reflection of g x b x in the line y = x. Graph of f x logb x for b > 1. Graph of f x logb x for 0 < b < 1. Note that the y-axis is a vertical asymptote for the graph of f x logb x . The domain of f x logb x is x > 0, and the range is all real numbers. Alg 2 Adv Ch. 7B Notes-page 3 Ex. 7: Graph logarithmic functions. Graph the function. a. y log 2 x b. y log 2 3 x * Translations: You can graph a logarithmic function of the form y logb x h k by translating the graph of the parent function y logb x . Ex.8: Translate a logarithmic graph. Graph. Then state the domain and range a. y log3 x 2 4 b. y log1 3 x 3 You Try: Graph. State the domain and range. a. y log5 x b. y log4 x 1 2 Alg 2 Adv Ch. 7B Notes-page 4 7-5 Apply Properties of Logarithms Name:__________________ Objective: To rewrite logarithmic expressions. Algebra 2 Standards 11.2, 13.0 and 14.0 *Properties of Logarithms: Let b, m and n be positive numbers such that b ≠1. Condensed form Expanded form Product Property If there is multiplication inside one log, you can split it up as addition of two logs. logb mn logb m logb n Quotient Property Power Property If there is division inside one log, you can split it up as subtraction of two logs. log b The logarithm of a power is equal to the product of the exponent and the logarithm. log b m n n log b m m logb m logb n n Ex. 1: Use Properties of logarithms. Use log3 12 2.262 and log3 2 0.631 to evaluate the logarithm. a. log3 6 b. log3 24 c. log3 32 You Try: Use log6 5 0.898 and log6 8 1.161 to evaluate the logarithm. 5 a. log 6 b. log6 40 c. log6 64 8 Ex. 2: Expand a logarithmic expression. 3x2 Ex. Expand log 7 3 5y Ex. 3: Which of the following is equivalent to ln8 2ln5 ln10 ? Alg 2 Adv Ch. 7B Notes-page 5 You Try: Expand log d. log6 125 x 2 y3 You Try: Condense ln 4 3ln3 ln12. *Change-of-Base Formula allows you to evaluate any logarithm using a calculator. For any positive real numbers, a, b, and x, a ≠ 1 and b ≠ 1, log a x log x ln x In particular, logb x log b x and logb x log a b log b ln b Ex. 4: Evaluate using common logarithms and natural logarithms. a. log6 24 b. log8 14 You Try: Use the Change-of-Base Formula to evaluate the logarithms. a. log5 8 b. log 26 9 c. log12 30 Ex. 5: The Richter scale is used to measure the magnitude of earthquakes. If an earthquake has I intensity, I, then its magnitude on the Richter scale, R, is given by the function R I log , I0 where I 0 is the intensity of a barely felt earthquake. If the intensity of one earthquake is 50 times that of another, how many points greater is the bigger earthquake on the Richter scale? Alg 2 Adv Ch. 7B Notes-page 6 7-6 Solve Exponential and Logarithmic Equations Name:__________________ Objective: To solve exponential and logarithmic equations. Algebra 2 Standards 11.1 and 11.2 *Exponential equations are equations in which variable expressions occur as exponents. *How to Solve Exponential Equations: 1st method: Write both sides using the same base if possible. (apply One-to-One Property of Exponential Functions) For b > 0 and b ≠ 1, bx = by if and only if x = y. Example: If 3x 35 , then x = 5. If x = 5, then 3x 35 . 2nd method: Take a logarithm of each side. Solve by equating exponents. 1 Ex. 1: 3 9 x 3 x Ex. 2: 2510 x 8 ( 1 42 x ) 125 Take a log of each side. (You need a calculator) Ex. 3: Solve 9 x 35 Ex. 4: 3e 2 x 16 5 You Try: 92 x 27 x1 You Try: 79 x 15 *Newton’s Law of Cooling: A cooling substance with an initial temperature, T0 , the temperature T after t minutes can be modeled by T T0 TR ert TR where TR is the surrounding temperature and r is the substance’s cooling rate. Ex. 5: Hot chocolate that has been heated to 90°C is poured into a mug and placed on a table in a room with a temperature of 20°C. If r = 0.145 when the time is measured in minutes, how long will it take for the hot chocolate to cool to a temperature of 30°C? Alg 2 Adv Ch. 7B Notes-page 7 *How to Solve Logarithmic Equations: 1st method: If the equation is “single log single log ” apply One-to-One Property of Logarithmic Functions. For b > 0 and b ≠ 1, logb x logb y if and only if x = y. Example: If log 2 x log 2 7 , then x = 7. 2nd method: If the equation is “single log a number ” rewrite in exponential form. (or exponentiate each side of an equation) Solve a logarithmic equation. Ex. 6: Solve log4 2 x 8 log4 6 x 12 You Try: ln 7 x 4 ln 2 x 11 Rewrite in exponential form (or exponentiate both sides) Ex. 7: Solve log7 3x 2 2 You Try: Solve log2 x 6 5 *Extraneous Solutions: Because the domain of a logarithmic function is x 0 if logb x , you need to check for extraneous solutions of any logarithmic equation. Example 8: What is (are) the solution(s) of log6 3x log6 x 4 2 ? A. B. C. D. -6, 2 -2, 6 2 6 Alg 2 Adv Ch. 7B Notes-page 8 You Try: Solve the equation. Check for extraneous solutions. a. log4 x 12 log4 x 3 b. log6 3x 10 log6 14 5x *Use a Logarithmic Model: Example 9: The population of deer in a forest preserve can be modeled by the equation P 50 200ln t 1 , where t is the time in years from the present. In how many years will the deer population reach 500? Example 10: Susie deposited $100 into an account that compounded interest quarterly. In 50 years, the account has a balance of $347.68. What is the interest rate? You Try: Find the amount of time required for an investment to triple at a rate of 10% if the interest is compounded continuously Alg 2 Adv Ch. 7B Notes-page 9 7-7 Write and Apply Exponential and Power Functions Objective: To write exponential and logarithmic equations. Algebra 2 Standards 12.0 Name:______________________ * How to write an exponential function (y = abx ) or a power function (y = axb ) whose graph passes through two points. 1) Substitute the given points into the function. 2) Solve for a and find b 3) Find a and determine the function. Ex. 1: Write an exponential function. Write an exponential function y = abx whose graph passes through (1, 10) and (4, 80). You Try: Write an exponential function y = abx whose graph passes through the points (1, 8) and (2, 32). Ex. 2: Use a calculator to find an exponential model The table shows the amount A in a savings account t years after the account was opened. 0 1 2 3 4 5 6 7 t A 210 255 310 377 459 557 677 822 Find an exponential model for the original data. Alg 2 Adv Ch. 7B Notes-page 10 You Try: Find an exponential model for the data: (0, 17.56), (1, 16.03), (2, 14.64), (3, 13.36), (4, 12.20), (5, 11.14), (6, 10.17) Ex. 3: Write a power function. Write a power function y = axb whose graph passes through (2, 4) and (6, 10). You Try: Write a power function whose graph passes through the points (3, 8) and (6,17). Ex.` 4: Find a power model. The table gives the approximate volume V of spheres with radius r. 1 2 3 4 5 6 r 4.189 33.510 113.097 268.083 V Find a power model for the original data. Alg 2 Adv Ch. 7B Notes-page 11 523.599 904.779