491 4.5 Exponential and Logarithmic Equations EXAMPLE 3 Solving Base e Exponential Equations Solve each equation. Give solutions to the nearest thousandth. (a) e x2 = 200 (b) e2x+1 # e-4x = 3e SOLUTION (a) e x2 = 200 ln e x2 = ln 200 Take the natural logarithm on each side. x2 = ln 200 ln e x2 = x2 x = {2ln 200 Square root property x ≈ {2.302 Use a calculator. The solution set is 5{2.3026. (b) e2x+1 # e-4x = 3e e-2x+1 = 3e am # an = am+n e-2x = 3 Divide by e; e-2x+1 e1 = e-2x+1-1 = e-2x ln e-2x = ln 3 Take the natural logarithm on each side. -2x ln e = ln 3 Power property -2x = ln 3 ln e = 1 x = - 1 2 ln 3 Multiply by - 1 2 . x ≈ -0.549 Use a calculator. The solution set is 5-0.5496. S Now Try Exercises 21 and 23. EXAMPLE 4 Solving an Exponential Equation (Quadratic in Form) Solve e2x - 4e x + 3 = 0. Give exact value(s) for x. SOLUTION If we substitute u = e x, we notice that the equation is quadratic in form. e2x - 4e x + 3 = 0 1e x22 - 4e x + 3 = 0 am n = 1an2m u2 - 4u + 3 = 0 Let u = e x. 1u - 121u - 32 = 0 Factor. u - 1 = 0 or u - 3 = 0 Zero-factor property u = 1 or u = 3 Solve for u. e x = 1 or e x = 3 Substitute e x for u. ln e x = ln 1 or ln e x = ln 3 Take the natural logarithm on each side. x = 0 or x = ln 3 ln e x = x; ln 1 = 0 Both values check, so the solution set is 50, ln 36. S Now Try Exercise 35. Remember both roots.
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