Questions: Solve for (x). [ log 3(9)-log 3(x)=6 x=square ]

Transcript text: Solve for $x$. \[ \begin{array}{l} \log _{3}(9)-\log _{3}(x)=6 \\ x=\square \end{array} \]

Solution

Solution Steps

To solve the equation $\log_{3}(9) - \log_{3}(x) = 6$, we can use the properties of logarithms. Specifically, we can use the quotient rule for logarithms, which states that $\log_{b}(a) - \log_{b}(c) = \log_{b}(\frac{a}{c})$. Applying this rule, we can rewrite the equation as $\log_{3}(\frac{9}{x}) = 6$. Then, we can convert the logarithmic equation to an exponential equation: $\frac{9}{x} = 3^6$. Finally, solve for $x$ by isolating it on one side of the equation.

Step 1: Rewrite the Logarithmic Equation

We start with the equation: \[ \log_{3}(9) - \log_{3}(x) = 6 \] Using the properties of logarithms, we can rewrite this as: \[ \log_{3}\left(\frac{9}{x}\right) = 6 \]

Step 2: Convert to Exponential Form

Next, we convert the logarithmic equation to its exponential form: \[ \frac{9}{x} = 3^6 \] Calculating $3^6$ gives us: \[ 3^6 = 729 \] Thus, we have: \[ \frac{9}{x} = 729 \]

Step 3: Solve for $x$

To isolate $x$, we rearrange the equation: \[ x = \frac{9}{729} \] Simplifying this fraction, we find: \[ x = \frac{1}{81} \]