453 4.2 Exponential Functions EXAMPLE 5 Solving an Exponential Equation Solve 2x+4 = 8x-6. SOLUTION Write each side of the equation using a common base. 2x+4 = 8x-6 2x+4 = 1232x-6 Write 8 as a power of 2. 2x+4 = 23x-18 1am2n = amn x + 4 = 3x - 18 Set exponents equal (Property (b)). -2x = -22 Subtract 3x and 4. x = 11 Divide by -2. Check by substituting 11 for x in the original equation. The solution set is 5116. S Now Try Exercise 81. EXAMPLE 6 Solving an Equation with a Fractional Exponent Solve x4/3 = 81. SOLUTION Notice that the variable is in the base rather than in the exponent. x4/3 = 81 A 2 3 x B 4 = 81 Radical notation for am/n 2 3 x = {3 Take fourth roots on each side. Remember to use {. x = {27 Cube each side. Check both solutions in the original equation. Both check, so the solution set is 5{276. Alternative Method There may be more than one way to solve an exponential equation, as shown here. x4/3 = 81 1x4/323 = 813 Cube each side. x4 = 13423 Write 81 as 34. x4 = 312 1am2n = amn x = {2 4 312 Take fourth roots on each side. x = {33 Simplify the radical. x = {27 Apply the exponent. The same solution set, 5{276, results. S Now Try Exercise 83. Later in this chapter, we describe a general method for solving exponential equations where the approach used in Examples 4 and 5 is not possible. For instance, the above method could not be used to solve an equation like 7x = 12 because it is not easy to express both sides as exponential expressions with the same base. In Example 6, we solve an equation that has the variable as the base of an exponential expression.
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