Through the use of theorems on logarithms, log ab+log bc+log cd-log(ay)/(dx) can be reduced to:

1957 AMC 12 Problem 5

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Through the use of theorems on logarithms,

\log\dfrac ab+\log\dfrac bc+\log\dfrac cd-\log\dfrac{ay}{dx}

can be reduced to:

\log\dfrac yx

\log\dfrac xy

1

0

\log\dfrac{a^2y}{d^2x}

Combine the first three logarithms by multiplying their arguments

The product

(\frac{a}{b})(\frac{b}{c})(\frac{c}{d})

telescopes to

\frac{a}{d}

Combining the logarithms gives

\begin{aligned} &\log\left(\frac ab\frac bc\frac cd\frac{dx}{ay}\right)\\ &\quad=\log\left(\frac ad\frac{dx}{ay}\right)\\ &\quad=\log\frac xy. \end{aligned}

Therefore, the correct answer is B.