Chapter 5: Exponential and Logarithmic Functions5.6: Solving Exponential Logarithmic EquationsDay 1Essential Question: Give examples of equations that can be solved by using the properties of exponents and logarithms.

5.6: Solving Exponential and Logarithmic EquationsPowers of the Same BaseSolve the equation 8x = 2x+18x = 2x+1(23)x = 2x+123x = 2x+1Set the exponents equal to each other3x = x+12x = 1x = 1/2

5.6: Solving Exponential and Logarithmic EquationsPowers of the Different BasesSolve the equation 5x = 25x = 2log52 = xlog 2/log 5 = xx = 0.4307

5.6: Solving Exponential and Logarithmic EquationsPowers of the Different BasesSolve the equation 24x-1 = 31-xTake one base and make it into a log problemlog231-x = 4x-1(1 x)log23 = 4x-1(1 x)(log 3/log 2) = 4x 1(1 x)(1.5850) = 4x 1Calculate log 3/log 21.5850 1.5850x = 4x 1Distribute on left2.5850 1.5850x = 4xAdd 1 to both sides2.5850 = 5.5850xAdd 1.5850x to both sidesx = 0.4628Divide by 5.5850

5.6: Solving Exponential and Logarithmic EquationsUsing SubstitutionSolve the equation ex e-x = 4ex e-x = 4Multiply all terms by ex to remove the negative exponente2x 1 = 4exSet everything equal to 0, substitute u = exe2x 4ex 1 = 0u2 4u 1 = 0This is now aQuadratic Equation

5.6: Solving Exponential and Logarithmic EquationsUsing SubstitutionSet u back to ex, and solve

5.6: Solving Exponential and Logarithmic EquationsAssignmentPage 386Problems 1-31, odd problemsShow work

Chapter 5: Exponential and Logarithmic Functions5.6: Solving Exponential Logarithmic EquationsDay 2Essential Question: Give examples of equations that can be solved by using the properties of exponents and logarithms.

5.6: Solving Exponential and Logarithmic EquationsApplications of Exponential EquationsRadiocarbon DatingThe half-life of carbon-14 is 5730 years, so the amount of carbon-14 remaining at time t is given by Many of these problems will deal with percentage of carbon-14 remaining, so P = 1 (i.e. 100%), and the amount remaining will be the percentage left.

5.6: Solving Exponential and Logarithmic EquationsApplications: Carbon DatingThe skeleton of a mastodon has lost 58% of its original carbon-14. When did the mastodon die?If 58% has been lost, then 42% remains

5.6: Solving Exponential and Logarithmic EquationsApplications: Compound InterestIf $3000 is to be invested at 8% per year, compounded quarterly, in how many years will the investment be wroth $10,680?

5.6: Solving Exponential and Logarithmic EquationsAssignmentPage 386Problems 53-67, odd problemsShow work

Chapter 5: Exponential and Logarithmic Functions5.6: Solving Exponential Logarithmic EquationsDay 3Essential Question: Give examples of equations that can be solved by using the properties of exponents and logarithms.

5.6: Solving Exponential and Logarithmic EquationsApplications: Population GrowthA culture started at 1000 bacteria. 7 hours later, there are 5000 bacteria. Find the function and when there are 1 billion bacteria.Function is based off A = Pert. Need to find r.

5.6: Solving Exponential and Logarithmic EquationsApplications: Population GrowthTo find A=1,000,000, need to find t

5.6: Solving Exponential and Logarithmic EquationsLogarithmic EquationsSolve the equation ln(x 3) + ln(2x + 1) = 2(ln x)ln[(x 3)(2x + 1)] = ln x2ln(2x2 5x 3) = ln x2Natural logs cancel each other out2x2 5x 3 = x2x2 5x 3 = 0Use quadratic equation

5.6: Solving Exponential and Logarithmic EquationsLogarithmic EquationsSolve the equation ln(x 3) + ln(2x + 1) = 2(ln x) Because = -0.5414, its undefined for ln(x 3), so theres only one solution

5.6: Solving Exponential and Logarithmic EquationsEquations with logarithmic & constant termsSolve ln(x 3) = 5 ln(x 3)ln(x 3) + ln(x 3) = 52 ln(x 3) = 5ln (x 3) = 2.5e2.5 = x 3e2.5 + 3 = xx = 15.1825

5.6: Solving Exponential and Logarithmic EquationsEquations with logarithmic & constant termsSolve log(x 16) = 2 log(x 1)log(x 16) + log(x 1) = 2log [(x 16)(x 1)] = 2log (x2 17x + 16) = 2102 = x2 17x + 160 = x2 17x 840 = (x 21)(x + 4)x = 21 or x = -4x = -4 would give log(-4 16) = log -20, which is undefinedThere is only one solution, x = 21

5.6: Solving Exponential and Logarithmic EquationsAssignmentPage 386Problems 35-51 & 69-75, odd problemsShow work