494 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions NOTE We could have replaced 1 with log10 10 in Example 8 by first writing log 13x + 22 + log 1x - 12 = 1 Equation from Example 8 log10 313x + 221x - 124 = log10 10 Substitute. 13x + 221x - 12 = 10, Property of logarithms and then continuing as shown on the preceding page. EXAMPLE 9 Solving a Base e Logarithmic Equation Solve ln eln x - ln 1x - 32 = ln 2. Give exact value(s). SOLUTION This logarithmic equation differs from those in Examples 7 and 8 because the expression on the right side involves a logarithm. ln eln x - ln 1x - 32 = ln 2 ln x - ln 1x - 32 = ln 2 eln x = x ln x x - 3 = ln 2 Quotient property x x - 3 = 2 Property of logarithms x = 21x - 32 Multiply by x - 3. x = 2x - 6 Distributive property x = 6 Solve for x. Check that the solution set is 566. S Now Try Exercise 79. Solving an Exponential or Logarithmic Equation To solve an exponential or logarithmic equation, change the given equation into one of the following forms, where a and b are real numbers, a 70 and a≠1, and follow the guidelines. 1. aƒ 1x2 =b Solve by taking logarithms on each side. 2. log a ƒ1x2 =b Solve by changing to exponential form ab = ƒ1x2. 3. log a ƒ1x2 =log a g1x2 The given equation is equivalent to the equation ƒ1x2 = g1x2. Solve algebraically. 4. In a more complicated equation, such as e2x+1 # e-4x = 3e, See Example 3(b). it may be necessary to first solve for aƒ1x2 or log a ƒ1x2 and then solve the resulting equation using one of the methods given above. 5. Check that each proposed solution is in the domain.
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