Solve the equation for \( x \). Give an exact solution if possible otherwise give an approximation to 3-decimal places. \( \log _{4}(x)-\log _{4}(x+6)=\log _{4}(x-5) \)
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To solve the equation \( \log_{4}(x) - \log_{4}(x+6) = \log_{4}(x-5) \), we can use the properties of logarithms. The left side simplifies using the quotient rule of logarithms: \[ \log_{4}\left(\frac{x}{x+6}\right) = \log_{4}(x-5) \] Since both sides are logarithms with the same base, we can set the arguments equal to each other: \[ \frac{x}{x+6} = x - 5 \] Now, cross-multiply to eliminate the fraction: \[ x = (x - 5)(x + 6) \] Expanding the right side gives: \[ x = x^2 + 6x - 5x - 30 \] Simplifying yields: \[ x = x^2 + x - 30 \] Rearranging the equation leads to: \[ 0 = x^2 + x - x - 30 \implies 0 = x^2 - 30 \] Factoring gives us: \[ 0 = (x - \sqrt{30})(x + \sqrt{30}) \] Thus, the solutions are \( x = \sqrt{30} \) and \( x = -\sqrt{30} \). Since \( x \) must be greater than 5 for the logarithm to be defined, the relevant solution is: \[ x = \sqrt{30} \approx 5.477 \] The exact solution is \( \sqrt{30} \), and the approximation to three decimal places is \( 5.477 \).