8.5 Solving Exponential Equations GOAL Solve exponential equations in one variable using a variety of strategies. LEARN ABOUT te Mat All radioactive substances decrease in mass over time. Jamie works in a laboratory tat uses radioactive substances. Te laboratory received a sipment of 200 g of radioactive radon, and days later, 12.5 g of te radon remained.? Wat is te alf-life of radon? EXAMPLE 1 Selecting a strategy to solve an exponential equation Calculate te alf-life of radon. Solution A: Solving te equation algebraically by writing bot sides wit te same base M(t) 5 P a 1 2 b t 12.5 5 200a 1 2 b 12.5 200 5 a1 2 b 1 5 a1 2 b a 1 2 b 4 5 a 1 2 b Write te formula for alf-life, were is te alf-life period. Substitute te known values M(t) (for te remaining mass), P (te original mass), and t (te time in days). Isolate by dividing bot sides of te equation by 200. Express te left side of te equation as a fraction. Bot sides can be written wit te same base. 480 8.5 Solving Exponential Equations
8.5 4 5 4 5 5 4 Te alf-life of radon is 4 days. Te powers are equal wen te exponents are equal. Solve te resulting equation. Multiply bot sides by, and ten divide by 4 to solve for. Solution B: Solving te equation algebraically by taking te logaritm of bot sides M(t) 5 P a 1 2 b t 12.5 5 200a 1 2 b 12.5 200 5 a1 2 b 0.0625 5 (0.5) log (0.0625) 5 log (0.5) log (0.0625) 5 log (0.5) log (0.0625) 5 log (0.5) log (0.5) 5 log (0.0625) 5 4 Write te formula for alf-life. Substitute te given values. Divide bot sides of te equation by 200. Express te fractions as decimals. If two quantities are equal, ten te logs of te quantities will also be equal. Use te power rule for logaritms to rewrite te rigt side of te equation witout an exponent. Multiply bot sides by. Divide bot sides of te equation by log (0.0625) and evaluate te result wit a calculator. Te alf-life of radon is 4 days. Capter 8 481
Solution C: Solving te equation grapically using graping tecnology M(t) 5 P a 1 2 b t 12.5 5 200a 1 2 b 12.5 200 5 a1 2 b Write te formula for alf-life. Substitute te given values. To solve for, divide bot sides of te equation by 200. 0.0625 5 (0.5) Express te fractions as decimals. Enter te rigt side of te equation into Y1 of te equation editor. Enter te left side into Y2. Grap using a window tat corresponds to te domain and range in tis context. Tec Support For elp using te graping calculator to determine points of intersection, see Tecnical Appendix, T-12. Determine te point of intersection of te graps using te intersect operation. Te x-coordinate of te point of intersection is te solution to te equation. Te alf-life of radon is 4 days. Reflecting A. Wy did te strategy tat was used in Solution A result in an exact answer? Will tis strategy always result in an exact answer? Explain. B. Wic of te strategies used in te tree different solutions will always result in an exact answer? Explain. C. Wic of te tree strategies do you prefer? Justify your preference. 482 8.5 Solving Exponential Equations
APPLY te Mat EXAMPLE 2 Selecting a strategy to solve an exponential equation wit more tan one power 8.5 Solve 2 x12 2 2 x 5 24. Solution 2 x12 2 2 x 5 24 2 x (2 2 2 1) 5 24 2 x (4 2 1) 5 24 2 x (3) 5 24 2 x 5 8 2 x 5 2 3 x 5 3 Te terms on te left side of te equation cannot be combined. Divide out te common factor of 2 x on te left side of te equation. Simplify te expression in brackets. Divide bot sides by 3. Express te rigt side of te equation as a power of 2. EXAMPLE 3 Using logaritms to solve a problem An investment of $2500 grows at a rate of 4.8% per year, compounded annually. How long will it take for te investment to be wort $4000? Recall tat te formula for compound interest is A 5 P(1 1 i) n. Solution A 5 P(1 1 i) n 4000 5 2500(1.048) n 4000 2500 5 (1.048)n 1.6 5 (1.048) n log (1.6) 5 log (1.048) n log (1.6) 5 n log (1.048) Substitute te known values (P 5 2500, i 5 0.048, and A 5 4000) into te formula. Te variable n represents te number of years. Divide bot sides of te equation by 2500. Express te result as a decimal. Take te log of bot sides to solve for n. Use te power rule for logaritms to rewrite te equation. n 5 log 1.6 log (1.048) 8 10.025 Divide bot sides of te equation by log (1.048) to solve for n. It will take approximately 10.025 years for te investment to be wort $4000. Capter 8 483
EXAMPLE 4 Solve 2 x11 5 3 x21 to tree decimal places. Solution Selecting a strategy to solve an exponential equation wit different bases 2 x11 5 3 x21 log (2 x11 ) 5 log (3 x21 ) (x 1 1) log 2 5 (x 2 1) log 3 x log 2 1 log 2 5 x log 3 2 log 3 log 2 1 log 3 5 x log 3 2 x log 2 log 2 1 log 3 x (log 3 2 log 2) 5 log 3 2 log 2 log 3 2 log 2 log 2 1 log 3 log 3 2 log 2 5 x 4.419 8 x Bot sides of te equation cannot be written wit te same base. Take te log of bot sides of te equation. Use te power rule for logaritms to rewrite bot sides of te equation wit no exponents. Expand using te distributive property. Collect like terms to solve te equation. Divide out te common factor of x on te rigt side. Ten divide bot sides by log 3 2 log 2. Evaluate using a calculator. Round te answer to te required number of decimal places. In Summary Key Ideas Two exponential expressions wit te same base are equal wen teir exponents are equal. For example, if a m 5 a n, ten m 5 n, were a. 0, a 2 1, and m, npr. If two expressions are equal, taking te log of bot expressions maintains teir equality. For example, if M 5 N, ten log a M 5 log a N, were M, N. 0, a. 0, a 2 1. Need to Know To solve an exponential equation algebraically, take te logaritm of bot sides of te equation using a base of 10, and ten use te power rule for logaritms to simplify te equation and solve for te unknown. Sometimes an exponential equation can be solved algebraically by writing bot sides of te equation wit te same base (if possible), setting te exponents equal to eac oter, and solving for te unknown. Exponential equations can also be solved wit graping tecnology, using te same strategies tat are used for oter kinds of equations. 484 8.5 Solving Exponential Equations
CHECK Your Understanding 1. Solve. a) 5 x 5 625 c) 9 x11 5 27 2x23 e) b) 4 2x 5 2 52x d) 8 x21 5! 3 f) 2. Solve. Round your answers to tree decimal places. a) 2 x 5 17 c) 30(5 x ) 5 150 e) b) 6 x 5 231 d) 210 5 40(1.5)x f) 3. Solve by rewriting in exponential form. a) x 5 log 3 243 c) x 5 log 55!5 e) b) x 5 log 6 2 d) x 5 log 2! 5 8 f ) 2 3x 5 1 2 4 2x 5 1 5 12x 5 10 6 x 3 5 30 x 5 log 2 a 1 4 b x 5 log 3 a 1!3 b 8.5 PRACTISING 4. Te formula to calculate te mass, M(t), remaining from an original sample of radioactive material wit mass P, is determined using te formula M(t) 5 P Q 1 t were t is time and is te alf-life of te 2 R, substance. Te alf-life of a radioactive substance is 8. How long will it take for a 300 g sample to decay to eac mass? a) 200 g b) 100 g c) 75 g d) 20 g 5. Solve. K a) 49 x21 5 7"7 d) b) 2 3x24 5 0.25 e) c) a 1 x14 4 b 5 "8 f) 36 2x14 5 Q"1296R x 2 2x12 1 7 5 71 9 2x11 5 81(27 x ) 6. a) If $500 is deposited into an account tat pays 8%>a compounded A annually, ow long will it take for te deposit to double? b) A $1000 investment is made in a trust fund tat pays 12%>a compounded montly. How long will it take te investment to grow to $5000? c) A $5000 investment is made in a savings account tat pays 10%>a compounded quarterly. How long will it take for te investment to grow to $7500? d) If you invested $500 in an account tat pays 12%>a compounded weekly, ow long would it take for your deposit to triple? 7. A bacteria culture doubles every 15 min. How long will it take for a culture of 20 bacteria to grow to a population of 3 840? Capter 8 485
8. Solve for x. a) 4 x11 1 4 x 5 0 d) b) 2 x12 1 2 x 5 320 e) c) 2 x12 2 2 x 5 96 f) 10 x11 2 10 x 5 9000 3 x12 1 3 x 5 30 4 x13 2 4 x 5 63 9. Coose a strategy to solve eac equation, and explain your coice. (Do not solve.) a) 225(1.05) x 5 450 b) 3 x12 1 3 x 5 270 10. Solve. Round your answers to tree decimal places. a) 5 t21 5 3.92 c) 4 2x 5 5 2x21 b) x 5 log 3 25 d) x 5 log 2 53.2 11. A plastic sun visor allows ligt to pass troug, but reduces te intensity of te ligt. Te intensity is reduced by 5% if te plastic is 1 mm tick. Eac additional millimetre of tickness reduces te intensity by anoter 5%. a) Use an equation to model te relation between te tickness of te plastic and te intensity of te ligt. b) How tick is a piece of plastic tat reduces te intensity of te ligt to 60%? 12. Solve 3 2x 2 5(3 x ) 526. T 13. If log, sow tat y 5 log x ax 5 y Explain ow tis relationsip could log a. C be used to grap y 5 log 5 x on a graping calculator. Extending 14. Solve for x. a) b) 3 x 2 120 5 a 1 3x 2 x 2 5 32(2 4x ) c) 27 b 2 3 3 x 5 7 3 5 x. 15. If log, sow tat a 3 a 2 5 log b 8 5 b.. Determine te point of intersection for te graps of y 5 3(5 2x ) and y 5 6(4 3x ). Round your answer to tree decimal places. 17. Solve for x, to two decimal places. a) 6 3x 5 4 2 x23 b) (1.2) x 5 (2.8) x14 c) 3(2) x 5 4 x11 18. Solve for x, to two decimal places. (2 x ) x 5 10 486 8.5 Solving Exponential Equations