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Name _________________________________________________________ Date __________ 6.6 Solving Exponential Equations For use with Exploration 6.6 Essential Question How can you solve an exponential equation graphically? 1 EXPLORATION: Solving an Exponential Equation Graphically Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use a graphing calculator to solve the exponential equation 2.5x−3 = 6.25 graphically. Describe your process and explain how you determined the solution. 2 EXPLORATION: The Number of Solutions of an Exponential Equation Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. x y = 2. a. Use a graphing calculator to graph the equation 6 −6 6 −2 b. In the same viewing window, graph a linear equation (if possible) that does not x y = 2. intersect the graph of c. In the same viewing window, graph a linear equation (if possible) that intersects the graph of y = 2x in more than one point. d. Is it possible for an exponential equation to have no solution? more than one solution? Explain your reasoning. Copyright © Big Ideas Learning, LLC Integrated Mathematics I 173 All rights reserved. CopyrStudent Jouight © Big Idears Lnalea rning, LLC All rights reserved. 213 Name _________________________________________________________ Date _________ 6.6 Solving Exponential Equations (continued) 3 EXPLORATION: Solving Exponential Equations Graphically Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use a graphing calculator to solve each equation. 2x = 1 x+1 x a. b. 20= c. 2 21= x x−1 42x = 1 d. 39= e. 30= f. 16 3x 1 x+2 1 x−2 3 2 = 3 = 22=−x g. 8 h. 9 i. 2 Communicate Your Answer 4. How can you solve an exponential equation graphically? 5. A population of 30 mice is expected to double each year. The number p of mice in the population each year is given by p = 30 2n . In how many years will there be () 960 mice in the population? Integrated Mathematics I Copyright © Big Ideas Learning, LLC 174 Copyright © Big Ideas Learning, LLC All rights reserved. Student Journal All rights reserved. 214 Name _________________________________________________________ Date __________ 6.6 Practice For use after Lesson 6.6 Core Concepts Property of Equality for Exponential Equations b ≠ 1, Words Two powers with the same positive base b, where are equal if and only if their exponents are equal. x 5 x = 5, x 5 Numbers If then x = 5. If then 2 = 2. 2 = 2, b ≠ 1, x y Algebra If and then bb= if and only if x = y. b > 0 Notes: Worked-Out Examples Example #1 Solve the equation. Check your solution. 9x 7x+8 9x 7x+8 3 = 3 Check: 3 = 3 9(4) ? 7(4)+8 9x = 7x + 8 3 = 3 ? 36 28+8 − 7x − 7x 3 = 3 36 36 2x = 8 3 = 3 2x 8 17 17 — 1.50095 × 10 = 1.50095 × 10 ✓ = — 2 2 x = 4 Th e solution is x = 4. Integrated Mathematics I Copyright © Big Ideas Learning, LLC 175 All rights reserved. Student Journal 215 Name _________________________________________________________ Date _________ 6.6 PracticePRACTI(CcoEnytinued) Example #2 Solve the equation. Check your solution. 1 x+1 −3x+3 — 36 = ) (216 Check: 1 x+1 x+1 2 −3x+3 — 1 (6) = ) −3x+3 ( 3 — 36 = ) 6 (216 2(−3x+3) −3 x+1 6 = (6 ) −3(3)+3? 1 3+1 36 = 2(−3x+3) −3(x+1) — 6 = 6 (216) 2(−3x + 3) = −3(x + 1) −9+3 ? 1 4 36 = — 2(−3x) + 2(3) = −3(x) − 3(1) (216) −6 ? 1 36 = −6x + 6 = −3x − 3 — 2164 + 6x + 6x ? 1 1 = — —— 6 = 3x − 3 366 2,176,782,336 + 3 + 3 1 1 —— ✓ = 9 = 3x ——2,176,782,336 2,176,782,336 9 3x — = — 3 3 3 = x Th e solution is x = 3. Practice A Extra Practice In Exercises 1–15, solve the equation. Check your solution. 4x 12 x+5 20 45x− 2x 1. 33= 2. 88= 3. 66= 6x−3 −+34x 2x+11 3−2x 4. 55= 5. 4 = 1024 6. 8 = 512 7−x x−2 6x−1 5x 7. 4 = 256 8. 49 = 343 9. 36 = 6 x−43x x+1 x 2x 21x+ 10. 9 = 81 11. 64 = 512 12. 6 = 36 Integrated Mathematics I Copyright © Big Ideas Learning, LLC 176 Copyright © Big Ideas Learning, LLC All rights reserved. Student Journal All rights reserved. 216
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