Question

Through the use of theorems on logarithms $log\frac{a}{b}+log\frac{b}{c}+log\frac{c}{d}-log\frac{ay}{dx}$ shall be reduced to:

• A. $log\frac{y}{x}$
• B. $log\frac{x}{y}$
• C. 1
• D. 0
• E. $log\frac{a^{2}y}{d^{2}x}$

Answer 1

The correct option is B $log\frac{x}{y}$

We know the properties that:

$log\frac{l}{m}=log\left(l\right)-log\left(m\right)$ and $log\left(lm\right)=log\left(l\right)+log\left(m\right)$............$\left[1\right]$.

Using these properties we can reduce the given function as:

$\Rightarrow log\frac{a}{b}+log\frac{b}{c}+log\frac{c}{d}-log\frac{ay}{dx}$

$=log\left(a\right)-log\left(b\right)+log\left(b\right)-log\left(c\right)+log\left(c\right)-log\left(d\right)-log\left(a\right)-log\left(y\right)+log\left(d\right)+log\left(x\right)$

$=log\left(x\right)-log\left(y\right)=log\frac{x}{y}$.