Questions: Identify the choice that best completes the statement or answers the question. Condense the expression 7(log x-log y) to the logarithm of a single term. Select one: A. log (x^7/sqrt[7]y) B. log (x^7/y) C. 7(log x-log y) D. log ((x/y)^7) E. log (7x/7y)

Solution Steps

To condense the expression \(7(\log x - \log y)\) to the logarithm of a single term, we can use the properties of logarithms. Specifically, the difference of logs can be rewritten as the log of a quotient, and a constant multiplier can be rewritten as an exponent. Therefore, \(7(\log x - \log y)\) can be rewritten as \(\log \left(\frac{x}{y}\right)^7\).

We start with the expression \(7(\log x - \log y)\). Using the properties of logarithms, we can express the difference of logs as a quotient: \[ \log x - \log y = \log \left(\frac{x}{y}\right) \] Thus, we can rewrite the original expression as: \[ 7(\log x - \log y) = 7 \log \left(\frac{x}{y}\right) \]

Next, we apply the power rule of logarithms, which states that \(a \log b = \log(b^a)\). Therefore, we can express our equation as: \[ 7 \log \left(\frac{x}{y}\right) = \log \left(\left(\frac{x}{y}\right)^7\right) \]

Combining the results, we find that: \[ 7(\log x - \log y) = \log \left(\frac{x^7}{y^7}\right) \]

Final Answer