Solve a logarithmic equation

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Solve a logarithmic equation EXAMPLE 4 Solve a logarithmic equation Solve log (4x – 7) = log (x + 5). 5 SOLUTION log (4x – 7) = log (x + 5). 5 Write original equation. 4x – 7 = x + 5 Property of equality for logarithmic equations 3x – 7 = 5 Subtract x from each side. 3x = 12 Add 7 to each side. x = 4 Divide each side by 3. The solution is 4. ANSWER

Solve a logarithmic equation EXAMPLE 4 Solve a logarithmic equation Check: Check the solution by substituting it into the original equation. (4x – 7) = (x – 5) log 5 Write original equation. (4 4 – 7) = (4 + 5) ? log 5 Substitute 4 for x. 9 = 9 log 5 Solution checks.

Exponentiate each side of an equation EXAMPLE 5 Exponentiate each side of an equation Solve (5x – 1)= 3 log 4 SOLUTION (5x – 1)= (5x – 1)= 3 log 4 Write original equation. 4log4(5x – 1) = 4 3 Exponentiate each side using base 4. b = x log b x 5x – 1 = 64 5x = 65 Add 1 to each side. x = 13 Divide each side by 5. The solution is 13. ANSWER

Exponentiate each side of an equation EXAMPLE 5 Exponentiate each side of an equation log 4 (5x – 1) = (5 13 – 1) = 64 Check: log 4 Because 4 = 64, 64= 3. 3

Standardized Test Practice EXAMPLE 6 Standardized Test Practice SOLUTION log 2x + log (x – 5) = 2 Write original equation. log [2x(x – 5)] = 2 Product property of logarithms 10 = 10 log 2 [2x(x – 5)] Exponentiate each side using base 10. 2x(x – 5) = 100 Distributive property

Standardized Test Practice EXAMPLE 6 Standardized Test Practice 2x – 10x = 100 2 b = x log b x 2x – 10x – 100 = 0 2 Write in standard form. x – 5x – 50 = 0 2 Divide each side by 2. (x – 10)(x + 5) = 0 Factor. x = 10 or x = – 5 Zero product property Check: Check the apparent solutions 10 and –5 using algebra or a graph. Algebra: Substitute 10 and –5 for x in the original equation.

EXAMPLE 6 Standardized Test Practice log 2x + log (x – 5) = 2 log 2x + log (x – 5) = 2 log (2 10) + log (10 – 5) = 2 log [2(–5)] + log (–5 – 5) = 2 log 20 + log 5 = 2 log (–10) + log (–10) = 2 log 100 = 2 Because log (–10) is not defined, –5 is not a solution. 2 = 2 So, 10 is a solution.

EXAMPLE 6 Standardized Test Practice Graph: Graph y = log 2x + log (x – 5) and y = 2 in the same coordinate plane. The graphs intersect only once, when x = 10. So, 10 is the only solution. The correct answer is C. ANSWER

Solve the equation. Check for extraneous solutions. GUIDED PRACTICE for Examples 4, 5 and 6 Solve the equation. Check for extraneous solutions. 7. ln (7x – 4) = ln (2x + 11) 9. log 5x + log (x – 1) = 2 SOLUTION 5 SOLUTION 3 8. log (x – 6) = 5 2 10. log (x + 12) + log x =3 4 SOLUTION 4 SOLUTION 38