To determine which of the given options is equivalent to the logarithmic expression [tex]\(\log_7 18\)[/tex], we can use the change-of-base formula for logarithms. The change-of-base formula states:
[tex]\[
\log_b(a) = \frac{\log_c(a)}{\log_c(b)}
\][/tex]
In this formula, [tex]\(a\)[/tex] is the argument of the logarithm, [tex]\(b\)[/tex] is the base of the logarithm, and [tex]\(c\)[/tex] is any new base that we choose to use for the logarithms in the numerator and denominator.
For the given expression [tex]\(\log_7 18\)[/tex], we want to find an equivalent expression using base 10 logarithms (common logarithms), as it is commonly used. Applying the change-of-base formula to [tex]\(\log_7 18\)[/tex], we get:
[tex]\[
\log_7 18 = \frac{\log_{10}(18)}{\log_{10}(7)}
\][/tex]
Next, let's match this result with the given choices to identify the correct equivalent expression.
The provided options are:
A. [tex]\(\frac{\log_{18} 10}{\log_7 10}\)[/tex]
B. [tex]\(\frac{\log_{11} 18}{\log_{10} 11}\)[/tex]
C. [tex]\(\frac{\log_{10} 7}{\log_{10} 18}\)[/tex]
D. [tex]\(\frac{\log_{10} 18}{\log_{10} 7}\)[/tex]
By comparing our derived expression [tex]\(\frac{\log_{10}(18)}{\log_{10}(7)}\)[/tex] with the options, we can see that option D matches perfectly:
[tex]\[
\frac{\log_{10} 18}{\log_{10} 7}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
