SECTION 5.6 Logarithmic and Exponential Equations 339 used to obtain exact solutions.When algebraic techniques cannot be used, a graphing utility can be used to obtain approximate solutions. You should pay particular attention to the form of equations for which exact solutions are obtained. Solving an Exponential Equation Solve: = 2 5 x Algebraic Solution Because 5 cannot be written as an integer power of 2 ( ) = = 2 4 and 2 8 2 3 , write the exponential equation as the equivalent logarithmic equation. = = = x 2 5 log 5 ln5 ln2 x 2 ↑ Change-of-Base Formula Alternatively, the equation = 2 5 x can be solved by taking the natural logarithm (or common logarithm) of each side. = = = = ≈ x x 2 5 ln2 ln5 ln2 ln5 ln5 ln2 2.322 x x The solution set is { } ln5 ln2 . Graphing Solution Graph = Y 2x 1 and = Y 5 2 using a TI-84 Plus CE, and determine the x-coordinate of the point of intersection. See Figure 49. The approximate solution, rounded to three decimal places, is 2.322. Now Work PROBLEM 47 Solving an Exponential Equation Solve: ⋅ = 8 3 5 x Algebraic Solution Isolate the exponential expression and then rewrite the statement as an equivalent logarithm. ⋅ = = 8 3 5 3 5 8 x x ( ) ( ) = = ≈ − x log 5 8 ln 5 8 ln3 0.428 3 The solution set is ( ) { } log 5 8 3 . Graphing Solution Graph = ⋅ y 8 3x 1 and = y 5 2 using Desmos, and determine the x-coordinate of the point of intersection. See Figure 50. The approximate solution, rounded to three decimal places, is −0.428. Figure 49 16 22 21 4 Y2 5 5 Y1 5 2 x EXAMPLE 4 = = M N M N If , then ln ln . = M r M ln ln r Exact solution Approximate solution Figure 50 EXAMPLE 5 Solve for 3 .x Exact solution Approximate solution
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