Questions: Solve the logarithmic equation. ln (x+12)+ln (x-4)=2 ln x x=□ (Simplify your answer. Type an integer or a fraction.)

Solution Steps

Using the logarithm property \( \ln a + \ln b = \ln(ab) \), combine the logarithms: \[ \ln((x+12)(x-4)) = 2 \ln x. \]

Using the property \( a \ln b = \ln(b^a) \), rewrite the right side: \[ \ln((x+12)(x-4)) = \ln(x^2). \]

Since \( \ln A = \ln B \) implies \( A = B \), set the arguments equal: \[ (x+12)(x-4) = x^2. \]

Expand the left side: \[ x^2 + 8x - 48 = x^2. \] Subtract \( x^2 \) from both sides: \[ 8x - 48 = 0. \]

Add 48 to both sides: \[ 8x = 48. \] Divide by 8: \[ x = 6. \]

Check that \( x = 6 \) satisfies the original equation: \[ \ln(6+12) + \ln(6-4) = \ln(18) + \ln(2) = \ln(36), \] and \[ 2 \ln 6 = \ln(6^2) = \ln(36). \] Since both sides are equal, \( x = 6 \) is valid.

Final Answer