EXPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4 EXPONENTIAL & LOG EQUATIONS In a) b) c) the previous sections, we covered: Definitions of logs and exponential functions Graphs of logs and exponential functions Properties of logs and exponential functions In this section, , we are going to study procedures for solving equations involving logs and exponential equations e.g. : e 2x 9e x 36 0 EXPONENTIAL & LOG EQUATIONS In the last section, we covered two basic properties, which will be key in solving exponential and log equations. 1. One-to-One Properties a) a a x y x 2. y b) log a x log a y x y Inverse Properties a) log a a x x b) a loga x x EXPONENTIAL & LOG EQUATIONS We can use these properties to solve simple equations: 2 x 32 x 2 2 5 x 2 x 1 9 3 ln x 3 3 3 e ln x e 3 x 5 x -2 x e -3 EXPONENTIAL & LOG EQUATIONS When solving exponential equations, there are two general keys to getting the right answer: 1. Isolate the exponential expression 2. Use the 2nd one-to-one property log a x log a y x y EXPONENTIAL & LOG EQUATIONS Solve the following equation: 4 x 72 Isolate the exponential expression: Apply the 2nd one-to-one property log 4 4 x log 4 72 log 72 3.085... x log 4 EXPONENTIAL & LOG EQUATIONS Solve the following equation: 3(2 ) 42 x Isolate the exponential expression: 2 x 14 Apply the 2nd one-to-one property log 2 2 x log 214 log 14 3.807... x log 2 EXPONENTIAL & LOG EQUATIONS Solve the following equation: 4(e ) 16 2x Isolate the exponential expression: e 2x 4 Apply the 2nd one-to-one property ln e ln 4 2x 2x ln 4 ln 4 x 0.693 2 EXPONENTIAL & LOG EQUATIONS Solve a) b) c) the following equations: e x 5 60 x ln 55 x 4.007 ln 4 2(3 1) 10 x ln 3 x 5(e x2 x 1.262 22 - 2 x - 0.518 ) 8 14 x ln 5 EXPONENTIAL & LOG EQUATIONS Solving Equations of the Quadratic Type Two or more exponential expressions Similar procedure to what we have been doing Algebra is more complicated EXPONENTIAL & LOG EQUATIONS Solve the following equation: e 2 x 3e x 2 0 Start by rewriting the equation in quadratic form. (e ) 3e 2 0 x 2 x Factor the quadratic equation: let x e x x 2 3 x 2 0 ( x 2)( x 1) 0 (e x 2)(e x 1) 0 EXPONENTIAL & LOG EQUATIONS (e 2)(e 1) 0 x x e 2 0 ex 1 0 e 2 ex 1 x x ln ex ln 2 x ln 2 x .693 x0 EXPONENTIAL & LOG EQUATIONS Solve the following equation: e 2 x e x 20 0 (e ) e 20 0 x 2 x e 5 0 x e 5 x ln 5 x 1.609 x (e x 5)(e x 4) 0 e 40 x e 4 x ln 4 x error x EXPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4 EXPONENTIAL & LOG EQUATIONS Solve the following equation: 8 3x 5 5 3 8 x x 5 log 3 3 log 3 8 x log 5 8 log 3 x 0.428 EXPONENTIAL & LOG EQUATIONS Solve the following equation: 5 x 2 33 x 2 Since these are exponential functions of a different base, start by taking the log of both sides log 5 x 2 log 33 x 2 ( x 2) log 5 (3x 2) log 3 EXPONENTIAL & LOG EQUATIONS ( x 2) log 5 (3x 2) log 3 x log 5 2 log 5 3 x log 3 2 log 3 x log 5 3x log 3 2 log 3 2 log 5 x(log 5 3 log 3) 2 log 3 2 log 5 2 log 3 2 log 5 x 3.212 log 5 3 log 3 EXPONENTIAL & LOG EQUATIONS So far, we have solved only exponential equations Today, we are going to study solving logarithmic equations Similar to solving exponential equations EXPONENTIAL & LOG EQUATIONS Just as with exponential equations, there are two basic ways to solve logarithmic equations 1) Isolate the logarithmic expression and then write the equation in equivalent exponential form 2) Get a single logarithmic expression with the same base on each side of the equation; then use the one-to-one property EXPONENTIAL & LOG EQUATIONS Solve the following equation: ln x 2 Isolate the log expression: Rewrite the expression in its equivalent exponential form e x 2 x 7.389 EXPONENTIAL & LOG EQUATIONS Solve the following equation. log 3 (5x 1) log 3 ( x 7) 0 Get a single log expression with the same base on each side of the equation, then use the one-to-one property log 3 (5 x 1) log 3 ( x 7) 5x 1 x 7 x2 EXPONENTIAL & LOG EQUATIONS Solve the following equation 2 log 5 3x 4 Isolate the log expression: log 5 3x 2 Rewrite the expression in exponential form 5 3x 25 x 3 2 EXPONENTIAL & LOG EQUATIONS In some problems, the answer you get may not be defined. Remember, Therefore, y log a x is only defined for x > 0 if you get an answer that would give you a negative “x”, the answer is considered an extraneous solution EXPONENTIAL & LOG EQUATIONS Solve the following equation log 10 5x log 10 ( x 1) 2 Isolate the log expression: log 10[5x( x 1)] 2 Rewrite the expression in exponential form 10 5 x( x 1) 2 EXPONENTIAL & LOG EQUATIONS 100 5 x 5 x 2 x x 20 0 2 ( x 5)( x 4) 0 x 4, 5 Would either of these give us an undefined logarithm? log 10 5x log 10 ( x 1) 2 EXPONENTIAL & LOG EQUATIONS Solve the following equations: x 10 5 x 316.228 a) 2 log x 5 b) ln x 2 ln x x 1, 2 c) log x log( x 3) 1 2 10 x 3 EXPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4 - Applications EXPONENTIAL & LOG EQUATIONS Solve the following equation: 2 log 5 x log 5 9 log 5 x 2 log 5 9 x2 9 x3 x 3 EXPONENTIAL & LOG EQUATIONS Solve the following equation: log 4 ( x 3) log 4 (2 x) 1 log 4 ( x 3)(2 x) 1 4 ( x 3)( 2 x) x 2 x 6 4 x x20 2 EXPONENTIAL & LOG EQUATIONS x2 x 2 0 x2 x 2 0 EXPONENTIAL & LOG EQUATIONS How long would it take for an investment to double if the interest were compounded continuously at 8%? What is the formula for continuously compounding interest? A Pert If you want the investment to double, what would A be? A 2P EXPONENTIAL & LOG EQUATIONS A Pert 2 P Pe0.08t 2 e 0.08t 0.08t ln 2 ln 2 ln e ln 2 t 0.08 It will take about 8.66 years to double. 0.08t EXPONENTIAL & LOG EQUATIONS You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double? EXPONENTIAL & LOG EQUATIONS You have $50,000 to invest. You need to have $350,000 to retire in thirty years. At what continuously compounded interest rate would you need to invest to reach your goal? EXPONENTIAL & LOG EQUATIONS For selected years from 1980 to 2000, the average salary for secondary teachers y (in thousands of dollars) for the year t can be modeled by the equation: y = -38.8 + 23.7 ln t Where t = 10 represents 1980. During which year did the average salary for teachers reach 2.5 times its 1980 level of $16.5 thousand? EXPONENTIAL & LOG EQUATIONS EXPONENTIAL & LOG EQUATIONS EXPONENTIAL & LOG EQUATIONS