Questions: Solve the logarithmic equation algebraically. log4 x - log4(x-9) = 1/2

Solution Steps

To solve the logarithmic equation \(\log _{4} x-\log _{4}(x-9)=\frac{1}{2}\), we can use the properties of logarithms. First, apply the quotient rule for logarithms to combine the left side into a single logarithm. Then, convert the logarithmic equation into an exponential equation to solve for \(x\).

We start with the equation: \[ \log_{4} x - \log_{4}(x - 9) = \frac{1}{2} \] Using the quotient rule for logarithms, we can combine the left side: \[ \log_{4} \left( \frac{x}{x - 9} \right) = \frac{1}{2} \]

Next, we convert the logarithmic equation to its exponential form: \[ \frac{x}{x - 9} = 4^{\frac{1}{2}} \] Since \(4^{\frac{1}{2}} = 2\), we have: \[ \frac{x}{x - 9} = 2 \]

Now, we can cross-multiply to solve for \(x\): \[ x = 2(x - 9) \] Expanding and rearranging gives: \[ x = 2x - 18 \implies x - 2x = -18 \implies -x = -18 \implies x = 18 \]

Final Answer