9.5 Exponential Equations & Inequalities

Logarithmic vocabulary Consider: log 260 Also: log 0.26 Ex 1) Underline the mantissa & circle the characteristic log 425 = 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 log 10 2 = = = log (2.6 × 10 –1 ) = log log 10 –1 = –1 = – mantissacharacteristic

Ex 2) Solve for x to the nearest hundredth. a) log x = = x = x b)2 log x = log x = = x = x c) ln x – 3 = ln x = e = x = 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)

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 =

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

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 = 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

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