Question

If $lo{g}_{12}18=a$ and $lo{g}_{24}54=b$, then $ab+5(a-b)$ is equal to

• A. $-1$
• B. $-2$
• C. $3$
• D. none of these

Answer 1

The correct option is A $-1$

Given,

$lo{g}_{12}18=a,lo{g}_{24}54=b$

So $a\frac{lo{g}_{2}18}{lo{g}_{2}12}=\frac{lo{g}_2(2\times 3^2)}{lo{g}_2(2^2\times 3)}=\frac{1+2lo{g}_{2}3}{2+lo{g}_{2}3}-\left(1\right)$

$b=\frac{log54}{log24}=\frac{lo{g}_2(2\times 3^3)}{lo{g}_2(2^3\times 3)}=\frac{1+3lo{g}_{2}3}{3+lo{g}_{2}3}-\left(2\right)$

Solving eqs.(1) and (2) we get,

$lo{g}_{2}3=\frac{2a-1}{2-a}=\frac{3b-1}{3-b}$

Equating a and b in above equaion

$\therefore (2a-1)(3-b)=(2-a)(3b-1)$

$6a-2ab-3+b=6b-2-3ab+a\to 6a-6b+b-a-2ab+3ab-3+2=0$

$\to 5a-5b+ab-1=0\to 5(a-b)+ab=-1$