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## 9.5 Exponential Equations & Inequalities

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**Logarithmic vocabulary**Consider: log 260 Also: log 0.26 Ex 1) Underline the mantissa & circle the characteristic log 425 = 2.6284 If we are given log x or lnx, we can find x using our calculators. We will use 10x or ex. Let’s practice some simple ones…get those calculators ready! = 2.4150 = log (2.6 × 102) = log 2.6 + log 102 = 0.4150 + 2 = –0.5850 = log (2.6 × 10–1) = log 2.6 + log 10–1 = 0.4150 + –1 mantissa characteristic**Ex 2) Solve for x to the nearest hundredth.**• a) log x = 3.2274 c) lnx – 3 = 5.7213 lnx = 8.7213 e8.7213 = x 6132.15 = x 2 log x = 2.6419 log x = 1.32095 101.32095 = x 20.94 = x 103.2274 = x 1688.11 = x • To solve an exponential equation: • Isolate the exponential expression • Take the logarithm of both sides of the equation • Verify all answers! (by substitution in original)**Ex 3) Solve for x to nearest hundredth.**42x – 1 – 27 = 0 42x – 1 = 27 log 42x – 1 = log 27 (2x – 1) log 4 = log 27 2x log 4 – log 4 = log 27 2x log 4 = log 27 + log 4 log 27 + log 4 2 log 4 x = 1.69 Ex 4) x =**Sometimes we can’t solve algebraically, so we go to our**graphing calculator. Solve using a graphing calculator. Ex 5) ex = x2 – 1 Y1 = ex Y2 = x2 – 1 (Find intersection) x = –1.15 Ex 6) y ≥ ex – 2 Y1 = ex – 2**Applications**Compound Interest Formula: A = total value of investment t = number of years P = principal amount invested r = interest rate n = number of times per year interest is compounded (% decimal)**Ex 7) The Smith Family wants to give their youngest**daughter $20,000 when she is ready for college. They now have $11,500 to invest. Determine how many years it will take them to achieve their goal given that they invest this amount at 8.3% compounded monthly. A = 20,000 P = 11,500 r = .083 n = 12 *Watch those parentheses! t = 7 years**Continuous Compound Interest Formula**A = Pert Ex 8) A sum of money invested at a fixed interest rate, compounded continuously, tripled in 19 years. Determine the interest rate at which the money was invested. *you don’t know A or P but you don’t need it! You need P to triple A = Pert 3P = Per(19) P P 3 = e19r ln 3 = 19rlne ln 3 19 = r r = 5.8%**Homework**#906 Pg 472 #1, 3, 5, 9, 13, 18, 20–23, 25, 27, 29, 32, 33, 36, 38, 39–47 odd