To solve the equation \( 6\log_x 2 - 3\log_x 3 = 3\log_5 0.2 \), we will break it down step by step. ### Step 1: Simplify the Right Side First, we need to simplify the right side of the equation, \( 3\log_5 0.2 \). We can rewrite \( 0.2 \) as \( \frac{1}{5} \): \[ \log_5 0.2 = \log_5 \left(\frac{1}{5}\right) = \log_5 5^{-1} = -1 \] Thus, we have: \[ 3\log_5 0.2 = 3 \times (-1) = -3 \] ### Step 2: Rewrite the Equation Now, substituting this back into the original equation gives us: \[ 6\log_x 2 - 3\log_x 3 = -3 \] ### Step 3: Rearranging the Equation Next, we can rearrange the equation to isolate the logarithmic terms: \[ 6\log_x 2 - 3\log_x 3 + 3 = 0 \] ### Step 4: Factor Out Common Terms We can factor out the common logarithmic terms: \[ 6\log_x 2 - 3\log_x 3 = 3 \] Dividing the entire equation by 3 gives: \[ 2\log_x 2 - \log_x 3 = 1 \] ### Step 5: Use Change of Base Formula Using the change of base formula, we can express the logarithms in terms of base 10 (or natural logarithm): \[ \log_x 2 = \frac{\log 2}{\log x} \quad \text{and} \quad \log_x 3 = \frac{\log 3}{\log x} \] Substituting these into our equation gives: \[ 2\left(\frac{\log 2}{\log x}\right) - \left(\frac{\log 3}{\log x}\right) = 1 \] ### Step 6: Combine the Terms Combining the terms over a common denominator: \[ \frac{2\log 2 - \log 3}{\log x} = 1 \] ### Step 7: Solve for \(\log x\) Now, we can multiply both sides by \(\log x\): \[ 2\log 2 - \log 3 = \log x \] ### Step 8: Exponentiate to Solve for \(x\) Exponentiating both sides gives us: \[ x = 10^{(2\log 2 - \log 3)} \] Using properties of logarithms, we can simplify this further: \[ x = 10^{\log(2^2) - \log 3} = 10^{\log\left(\frac{4}{3}\right)} = \frac{4}{3} \] ### Conclusion Thus, the value of \(x\) is: \[ \boxed{\frac{4}{3}} \] ### Explanation of Other Options - **Option A: \( \frac{8}{3} \)**: This value is greater than \( \frac{4}{3} \) and does not satisfy the logarithmic equation derived. - **Option C: \( \frac{3}{4} \)**: This value is less than \( \frac{4}{3} \) and does not satisfy the logarithmic equation derived. - **Option D: \( \frac{3}{8} \)**: This value is also less than \( \frac{4}{3} \) and does not satisfy the logarithmic equation derived. ### Revision Summary - Use the change of base formula to convert logarithms to a common base. - Rearrange logarithmic equations to isolate terms. - Combine logarithmic terms over a common denominator. - Exponentiate to solve for the variable when dealing with logarithmic equations.