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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 ln x, we can find x using our calculators. We will use 10 x or e x. Let’s practice some simple ones…get those calculators ready! = log (2.6 × 10 2 ) = log 2.6 + log 10 2 = 0.4150 + 2 = 2.4150 = log (2.6 × 10 –1 ) = log 2.6 + log 10 –1 = 0.4150 + –1 = –0.5850 mantissacharacteristic

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Ex 2) Solve for x to the nearest hundredth. a) log x = 3.2274 10 3.2274 = x 1688.11 = x b)2 log x = 2.6419 log x = 1.32095 10 1.32095 = x 20.94 = x c) ln x – 3 = 5.7213 ln x = 8.7213 e 8.7213 = x 6132.15 = x To solve an exponential equation: (1)Isolate the exponential expression (2)Take the logarithm of both sides of the equation (3)Verify all answers! (by substitution in original)

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Ex 3) Solve for x to nearest hundredth. 4 2x – 1 – 27 = 0 4 2x – 1 = 27 log 4 2x – 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 =

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Sometimes we can’t solve algebraically, so we go to our graphing calculator. Solve using a graphing calculator. Ex 5) e x = x 2 – 1 Y 1 = e x Y 2 = x 2 – 1 (Find intersection) x = –1.15 Ex 6) y ≥ e x – 2 Y 1 = e x – 2

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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)

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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

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Continuous Compound Interest Formula A = Pe rt 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. A = Pe rt 3P = Pe r(19) P P 3 = e 19r ln 3 = 19r lne ln 3 19 r = 5.8% *you don’t know A or P but you don’t need it! You need P to triple = r

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Homework #906 Pg 472 #1, 3, 5, 9, 13, 18, 20–23, 25, 27, 29, 32, 33, 36, 38, 39–47 odd

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If m & n are positive AND m = n, then Can solve exponential equation by taking logarithm of each side of equation Only works with base 10.

If m & n are positive AND m = n, then Can solve exponential equation by taking logarithm of each side of equation Only works with base 10.

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