Question

If $a^{3-x}.b^{5x}=a^{x+5}.b^{3x}$, show that $x\log \left(\frac{b}{a}\right)=\log a$.

Answer 1

$a^{3-x}{b}^{5x}=a^{x+5}{b}^{3x}$

Apply logarithm on both sides

$⟹(3-x)\log a+5x\log b=(x+5)\log a+3x\log b⟹3\log a-x\log a+5x\log b=x\log a+5\log a+3x\log b⟹x\left(2\log b-2\log a\right)=2\log a⟹x\log \left(\frac{b}{a}\right)=\log a$

Hence Proved