Question

Find the roots of the equation $2^{x+2}{.3}^{\frac{3x}{x-1}}=9$.

Answer 1

Given,

$2^{x+2}{.3}^{\frac{3x}{x-1}}=9$

Now taking both sides logarithm we get,

$(x+2)\log 2+\frac{3x}{x-1}\log 3=2\log 3$

or, $(x+2)\log 2+\frac{x+2}{x-1}\log 3=0$

or, $x+2=0$ and $\log 2+\frac{1}{x-1}\log 3=0$

or, $x=-2$ and $x=1-\frac{\log 3}{\log 2}$.