Question

If $a^p=b^q=c^r=d^s$ then ${\log }_a\left(bcd\right)$ is equal to

• A. $p(\frac{1}{q}+\frac{1}{r}+\frac{1}{s})$
• B. $1$
• C. $\frac{1}{q}+\frac{1}{r}+\frac{1}{s}$
• D. $0$

Answer 1

Answer: (A)

$p(\frac{1}{q}+\frac{1}{r}+\frac{1}{s})$

$a^p=b^q=c^r=d^3=k$ (say)

$\Rightarrow {\log }_a{a}^p={\log }_{a}k\Rightarrow p={\log }_{a}k$

Similarly, $q={\log }_{b}k;r={\log }_{c}k$ and $s={\log }_{d}k$

Now, ${\log }_a\left(bcd\right)=\frac{{\log }_{k}bcd}{{\log }_{k}a}$

$=\frac{{\log }_{k}b+{\log }_{k}c+{\log }_{k}d}{{\log }_{k}a}$

$=\frac{\frac{1}{q}+\frac{1}{r}+\frac{1}{s}}{\frac{1}{p}}=p(\frac{1}{q}+\frac{1}{r}+\frac{1}{s})$