Question

If ${\log }_{4}5=a\text{and lo}{\text{g}}_{5}6=b,\text{then lo}{\text{g}}_{3}2\text{is equal to}$

• A. $\frac{1}{2a+1}$
• B. $\frac{1}{2b+1}$
• C. $2ab+1$
• D. $\frac{1}{2ab-1}$

Answer 1

Answer: (D)

$\frac{1}{2ab-1}$

Given

${\log }_{4}5=a\Rightarrow \frac{\log 5}{\log 4}=a....................\left(1\right)$

${\log }_{5}6=b\Rightarrow \frac{\log 6}{\log 5}=b....................\left(2\right)$

Multiplying equation $\left(1\right)$ and $\left(2\right)$, we get:

${\log }_{4}6=ab$

$\Rightarrow {\log }_4(3\times 2)=ab$

$\Rightarrow {\log }_{2^2}\left(3\right)+{\log }_{2^2}2=ab$

$\Rightarrow \frac{1}{2}({\log }_{2}3+{\log }_{2}2)=ab$

$\Rightarrow ({\log }_{2}3+1)=2ab$

$\Rightarrow {\log }_{2}3=2ab-1$

$\Rightarrow {\log }_{3}2=\frac{1}{2ab-1}$ ...............[since ${\log }_{a}b=x$, then ${\log }_{b}a=\frac{1}{x}$]