MAT 150 – Class #19 Objectives Solve an exponential equation by writing it in logarithmic form Convert logarithms using the change of base formula Solve an exponential equation by using properties of logarithms Solve logarithmic equations Solve exponential and logarithmic inequalities Solving Exponential Equations Using Logarithmic Forms To solve an exponential equation using logarithmic form: 1. Rewrite the equation with the term containing the exponent by itself on one side. 2. Divide both sides by the coefficient of the term containing the exponent. 3. Change the new equation to logarithmic form. 4. Solve for the variable. Example Solve the equation 3000 150(10 ) for t by converting it to logarithmic form and graphically to confirm the solution. 4t Solution Divide both sides of the equation by 150. 4t 3000 150(10 ) 20 104t Rewrite in logarithmic form. 4t log20 Solve for t. log20 t 0.32526 4 Example (cont) Solve the equation 3000 150(10 ) for t by converting it to logarithmic form and graphically to confirm the solution. 4t Solution To solve graphically enter 3000 for y1 and 150(104t ) for y2 Example a. Prove that the time it takes for an investment to double its value is t ln 2 if the interest rate is r, r compounded continuously. Solution a. S Pert 2𝑃 = 𝑃𝑒 𝑟𝑡 2 = 𝑒 𝑟𝑡 log 𝑒 2 = 𝑟𝑡 ln 2 = 𝑟𝑡 ln2 t r Example (cont) b. Suppose $2500 is invested in an account earning 6% annual interest, compounded continuously. How long will it take for the amount to grow to $5000? Solution b. ln2 t r ln2 t 11.5525 0.06 Change of Base We can use a special formula called the change of base formula to rewrite logarithms so that the base is 10 or e. The general change of base formula is summarized below. Example Evaluate log8 124. Solution log124 log8 124 2.318 approximately log8 Example (cont) b. Graph the function by changing each logarithm to a common logarithm and then by changing the logarithm to a natural logarithm. y log3 x Solution change to base 10 y log3 x log x log3 change to base e= ln 𝑥 ln 3 Example If $10,000 is invested for t years at 10%, compounded annually, the future value is given by t S 10,000(1.10 ) In how many years will the investment grow to $45,950? Solution 45,950 10,000(1.10t ) 4.5950 1.10t t log1.10 4.5950 log4.5950 t log1.10 4.5950 16 log1.10 The investment will grow to $45,950 in 16 years. SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMIC PROPERTIES To solve an exponential equation using logarithmic properties: 1. Rewrite the equation with a base raised to a power on one side. 2. Take the logarithm, base e or 10, of both sides of the equation. 3. Use a logarithmic property to remove the variable from the exponent. 4. Solve for the variable. Example Solve the following exponential equations. a. 4096 82 x b. 6(43 x 2 ) 120 Solution a. Take log of base 10 of both sides. 4096 82 x log4096 log82 x Using the Power Property of Logarithms log4096 2 x log8 Solving for x log 4096 x 2log8 2x Example (cont) Solution b. 6(43 x2 ) 120 3 x 2 6(4 6 ) 120 6 43 x 2 20 ln 43 x 2 ln20 (3 x 2)ln 4 ln20 ln20 3x 2 ln 4 1 ln20 x 2 3 ln 4 x 1.387 Example Solve 4log3 x 8 by converting to exponential form and verify the solution graphically. Solution Divide both sides by 4: 4log3 x 8 log3 x 2 Write in exponential form: 3 2 x 1 x 9 Example Solve 6 3ln x 12 by converting to exponential form and verify the solution graphically. Solution 6 3ln x 12 3ln x 6 ln x 2 x e2 Example Solve ln x 3 ln( x 4) by converting to exponential form and then using algebraic methods. Solution ln x 3 ln( x 4) e3 x x 4 3 ln( x 4) ln x x4 3 ln x x4 3 e x e3 x x 4 x(e3 1) 4 4 x 3 0.21 e 1 Example After the end of an advertising campaign, the daily sales of Genapet fell rapidly, with daily sales given by S = 3200e-0.08x dollars, where x is the number of days from the end of the campaign. For how many days after the campaign ended were sales at least $1980? Solution 3200e 0.08 x 1980 e 0.08 x 0.61875 0.08 x ln e ln0.61875 0.08x 0.4801 x 6 Assignment Pg. 350-354 #3-7 odd #15-33 odd #47 #66