Question

If a=$lo{g}_{12}18$, b=$lo{g}_{24}54$ then the value of ab+5(a-b) is

• A. \N
• B. 4
• C. 1
• D. None of these

Answer 1

The correct option is C 1

We have

a=$lo{g}_{12}18=\frac{lo{g}_{2}18}{lo{g}_{2}12}=\frac{1+2lo{g}_{2}3}{2+lo{g}_{2}3}$ and

b=$lo{g}_{24}54=\frac{lo{g}_{2}54}{lo{g}_{2}24}=\frac{1+3lo{g}_{2}3}{3+lo{g}_{2}3}$.

Putting $x=lo{g}_{2}3,$ we have$ab+5(a-b)=\frac{1+2x}{2+x}.\frac{1+3x}{3+x}+(\frac{1+2x}{2+x}-\frac{1+3x}{3+x})=\frac{6x^2+5x+1+5(-x^2+1)}{(x+2)(x+3)}=\frac{x^2+5x+6}{(x+2)(x+3)}$