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Exponential and Logarithmic Equations Objectives: 1.Solve simple exponential and logarithmic equations 2.Solve more complicated exponential equations 3.Solve more complicated logarithmic equations 4.Use exponential and logarithmic equations to model and solve real-life applications WHY??? Applications of exponential and logarithmic equations are found in consumer safety testing. For example, a logarithmic function can be used to model crumple zones for automobile crash test Properties • One-to-One Properties a x a y if and only if x = y log a x log a y if and only if x = y • Inverse Properties a log a x x log a a x x Strategies for Solving Exponential and Logarithmic Equations • Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions • Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions • Rewrite a logarithmic equation in exponential form an apply the Inverse Property of exponential functions If au = av, then u = v This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal. The left hand side is 2 to the something. Can we re- 2 x 34 8 write the right hand side as 2 to the something? Now we use the property above. The bases are both 2 2 2x 34 3so the exponents must be equal. 3 x 34 We did not cancel the 2’s, We just used the property and equated the exponents. You could solve this for x now. 1 4 x The left hand side is 4 to the Let’s try one more: something but the right hand We could however re-write both the 8 side can’t be written as 4 to the something (using integer left and right hand sides as 2 to the exponents) something. 2 2 2x 3 So now that each side is written with the 2 same base we know the exponents must be x 2 3 2 equal. 3 x 23 Check: 4 1 2 3 1 1 1 8 1 x 4 8 3 2 2 8 2 3 4 EXAMPLES: Solving Simple Equations 2 64 x ln x ln 5 0 x 1 9 3 e 12 x ln x 4 log10 x 2 Steps for Solving an Exponential Equation Solve for the variable. If x is in more than one term get x terms on one side and factor out the x Use the 3rd Property of Logs to move the exponent out in front If you have au = bv and you can’t express a and b with the same base, take the log of both sides (ln or log) 25 x1 2 ln ln 5 x 23 1 x x 23 ln ln 12 5 x x2 3 5 n x 5 l 23 2ll l2x nn n Solve for the variable. If x is in more than one term get x terms on one side and factor out the x Use the 3rd Property of Logs to move the exponent out in front If you have au = bv and you can’t express a and b with the same base, take the log of both sides (ln or log) 2 5x1 x 23 5 n x 5 l 23 2ll l2 n xnn 3 n lx xl 2 2l l 2 n n5 5 n ln 2 2 2 ln 53 ln x ln 5 2 ln 2ln 5 52 ln 2 ln put in calculator 3 2 5 lnln making sure to x 18 . 2 enclose the 2 2 5 lnln numerator in parenthesis and Solve for the variable. If than one Solve for the variable. If x is in morex is in the denominator in term get x terms on one side and factor out the parenthesis more than one term get x terms on x one side and factor out the x Solving Exponential Equations 4 72 x 3 729 x 3(2 ) 42 x 2(5 ) 32 x If you have an exponential equation with a base “e” then isolate the e meaning get rid of other terms and coefficients and then take the ln of both sides. Remember that e’s and ln’s are inverses and “undo” each other so they’ll cancel out and you can solve from there. 3 6 x 2 e 2 1 isolate the “e” 3 e 4 x 21 2x 4 e 1 take the ln of both sides 3 4 2x 1 ln 2 4 3 ln e x1 ln 4 3 ln 1 x 3 .644 2 Solving an Exponential Equation e 5 60 x e 50 2x 23 2t5 4 11 84 62x 13 41 Solving an Exponential Equation of Quadratic Type Hint: Factor e 3e 2 0 2x x e 5e 6 0 2x x e 9e 36 0 2x x Steps for Solving a Logarithmic Equation CHECK! to make sure your answer is “legal” Solve for the variable. If x is in more than one term get x terms on one side and factor out the x Re-write the log equation in exponential form If the log is in more than one term, use log properties to condense The secret to solving log equations is to re-write the log equation in exponential form and then solve. 2 Convert 2x 1 log3 this to exponential form check: 3 2 21 x 7 3 log 1 22 2 8x 21 72x 3 2 8 log 7 x This is true since 23 = 8 2 4 l x oo g g1 l x4 3 use the first property of logs to “condense” 4x x3 log 1 under one log log a 4 x log MN 1 a log M aN x3 Re-write the log equation in exponential form If the log is in more than one term, use log properties to condense 4 l x oo g 1 3 g1 l x4 3 4 x x 3 4x 3 0 4 x xx 2 2 x x1 4 1 x, x 0 4 Solve for the variable. If x is in more than one term get x terms on one side and factor out the x Re-write the log equation in exponential form If the log is in more than one term, use log properties to condense 4 l oo g 4 x1 l x4 3 x, x g1 l 4 Remember that the domain of logs is lg 1 oo g 44 3 4 numbers greater than 0 so we need to make not “illegal” since first log is of 4 and sure that if we put our second is of 1 answers back in for x we won’t be trying to take the log of 0 or a 3 log 1 1 log 1 4 4 negative number. ah oh---can’t take the log of - 1 or - 4 so must throw this solution out CHECK! to make sure your answer is “legal” Solving a Logarithmic Equation ln x 2 ln x 7 log 3 (5x 1) log 3 (x 7) log10 (x 6) log10 (2x 1) Solving a Logarithmic Equation 5 2ln x 4 2 6ln x 10 2log 5 3x 4 log10 3z 2 Checking for Extraneous Solutions • Make sure you check your answers because not all results will be solutions to the equation log10 5x log10 (x 1) 2 Application #1 • You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take to double? A Pe rt Application #2 • Determine the amount of time it would take $1000 to double in an account that pays 6.75% interest, compounded continuously. How does this compare to “application #1”? Application #3 • The effective yield of a savings plan is the percent increase in the balance after 1 year. Find the effective yield for each savings plan when $1000 is deposited in a savings account. • 7% annual interest rate, compounded annually • 7% annual interest rate, compounded continuously • 7% annual interest rate compounded quarterly • 7.25% annual interest rate, compounded quarterly Which savings plan has the highest effective yield? Which savings plan will have the highest balance after 5 years? Application #4 • 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 ( 10 ≤ t ≤ 30) y 38.8 23.7ln t where t = 10 represents 1980. During which year did the average salary for secondary teachers reach 2.5 times its 1980 level of $16.5 thousand?
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