2.04 ExpLog - Exponential Equations Inequalities

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High School Physics
Academic year:

2024
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2 Exponential Functions 2.04 EQUATIONS AND INEQUALITIES Exponential Functions FINDING THE ZEROS Exponential Functions 1) Express both sides of equation using SAME BASE 2) Set exponents equal {if 𝑐 𝑚 = 𝑐 𝑛 , 𝑡ℎ𝑒𝑛 𝑚 = 𝑛} 3) Solve Finding the Zero 𝑓 𝑥 =3 4 0=3 4 𝑥−2 𝑥−2 − 192 192 = 3 4 64 = 4 − 192 𝑥−2 𝑥−2 1) Express both sides of equation using SAME BASE 43 = 4𝑥−2 2) Put Exponents equal {for 𝑐 > 0, 𝑐 ≠ 0, 𝑐 𝑚 = 𝑐 𝑛 → 𝑚 = 𝑛} 3=𝑥−2 3) Solve 5=𝑥 Ex 1: 1 𝑓 𝑥 = 625 5 1 0 = 625 5 Ex 2: −1 3𝑥 1 1 = 625 5 1 1 = 625 5 3𝑥 −1 3𝑥 3𝑥 5 𝑡𝑜 𝑤ℎ𝑎𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑒𝑞𝑢𝑎𝑙𝑠 625? 5 𝑡𝑜 𝑤ℎ𝑎𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑒𝑞𝑢𝑎𝑙𝑠 625? 1 1 = 625 5 3𝑥 1) Express both sides of equation using SAME BASE 4 3𝑥 1 1 = 5 5 2) Put Exponents equal {for 𝑐 > 0, 𝑐 ≠ 0, 𝑐 𝑚 = 𝑐 𝑛 → 𝑚 = 𝑛} 4 = 3𝑥 3) Solve 4 3 =𝑥 𝑓 𝑥 = 2𝑥−3 + 4 0 = 2𝑥−3 + 4 −4 = 2𝑥−3 𝐼𝑡 𝑖𝑠 𝑖𝑚𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑡𝑜 ℎ𝑎𝑣𝑒 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑝𝑜𝑤𝑒𝑟 Therefore there are no zeros Ex 3: Equals constant Exponential Functions 4=2 2 𝑥−4 64 = 2 2 32 = 2 − 60 𝑥−4 𝑥−4 25 = 2𝑥−4 5=𝑥−4 𝑥=9 • ∴ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑥 = {9} Ex 1: • 𝑏𝑟𝑖𝑛𝑔 − 60 𝑎𝑐𝑟𝑜𝑠𝑠 • 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 2 • 𝑚𝑎𝑘𝑒 𝑏𝑎𝑠𝑒𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 − 2 𝑡𝑜 𝑤ℎ𝑎𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 = 32 • 𝑠𝑒𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑒𝑞𝑢𝑎𝑙 • 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 • Check it on your calculator!! • • • • Make the bases the same Set exponents equal Solve for 𝑥 Check it on your calculator!! 1 4 1 4 1 4 1 4 Ex 2: 8𝑥 = 2 8𝑥 8𝑥 8𝑥 1 = 2 1 = 2 −10𝑥+18 10𝑥−18 2(5𝑥−9) (5𝑥−9) 1 = 4 8𝑥 = 5𝑥 − 9 3𝑥 = −9 𝑥 = −3 • ∴ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑥 = {−3} 𝑥 1 • make the bases the 3 = 9 same 32𝑥+8 = 3−2 𝑥 • 𝑠𝑒𝑡 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑒𝑞𝑢𝑎𝑙 32𝑥+8 = 3−2𝑥 • 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 2𝑥 + 8 = −2𝑥 • Check it on your 4𝑥 = −8 calculator!! 𝑥 = −2 • ∴ 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑥 = {−2} 𝑥+4 2 Ex 4: • Steps: 1.Sketch the graph 2.Solve for Equation (intersection point) 3.Evaluate Inequalities • 2 3 𝑥+5 − 62 < 100 • Solve for Equation 2 3 𝑥+5 − 62 = 100 2 3 𝑥+5 = 162 3 𝑥+5 = 81 • 3 to what exponent = 81? (4) 3 𝑥+5 = 34 • Set the exponents equal to each other Ex 1: 𝑥+5=4 𝑥 = −1 ] − ∞, 𝑃𝑂𝐼[ ∴ 𝑥 ∈] − ∞, −1[ Ex 1: • 2 1 −2𝑥+4 2 ≤ 2−𝑥+5 • Solve for Equation −2𝑥+4 1 2 = 2−𝑥+5 2 21 2−1 −2𝑥+4 = 2−𝑥+5 21 2 2𝑥−4 = 2−𝑥+5 22𝑥−4+1 = 2−𝑥+5 22𝑥−3 = 2−𝑥+5 • Set the exponents equal to each other Ex 2: 2𝑥 − 3 = −𝑥 + 5 3𝑥 = 8 8 𝑥= 3 •2 1 −2𝑥+4 2 ≤ 2−𝑥+5 −2𝑥+4 8 𝑥= 3 1 𝑓1 𝑥 = 2 2 𝑓2 𝑥 = 2−𝑥+5 𝑓1 𝑥 ≤ 𝑓2 𝑥 𝑜𝑣𝑒𝑟 ] − ∞, 𝑃𝑂𝐼] Ex 2: 8 ∴ 𝑥 ∈] − ∞, ] 3