Question

If $a^b=b^c=ab$, then $b+c$ always equals?

• A. $\frac{1}{bc}$
• B. $\frac{1}{2}bc$
• C. $1$
• D. $bc$

Answer 1

Answer: (D)

$bc$

$a^b=b^c=ab=p\left(let\right)$

$⟹a^b=p$

taking log

$⟹b\log a=\log p-\left(i\right)$

$b^c=p$

Taking log

$⟹c\log b=\log p-\left(ii\right)$

$ab=p$

Taking log

$\log a+\log b=\log p$

Using$\left(i\right)\&\left(ii\right),$we can say:

$\frac{\log p}{b}+\frac{\log p}{c}=\log p$

$⟹\frac{1}{b}+\frac{1}{c}=1$

$⟹b+c=bc$