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