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algebra intermediate

Problem

Through the use of theorems on logarithms

lo g \frac{a}{b} + lo g \frac{b}{c} + lo g \frac{c}{d} - lo g \frac{a y}{d x}

can be reduced to:

(A)

lo g \frac{y}{x}

(B)

lo g \frac{x}{y}

(C)

1

(D)

140 x - 24 x^{2} + x^{3}

Solution

Using the properties

lo g (x) + lo g (y) = lo g (x y)

and

lo g (x) - lo g (y) = lo g (x / y)

, we have

lo g \frac{a}{b} + lo g \frac{b}{c} + lo g \frac{c}{d} - lo g \frac{a y}{d x} = lo g (\frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d}) - lo g \frac{a y}{d x} = lo g \frac{a}{d} - lo g \frac{a y}{d x} = lo g (\frac{\frac{a}{d}}{\frac{a y}{d x}}) = lo g \frac{x}{y},

so the answer is

lo g \frac{x}{y} .

Final answer

B