Question-9

If a real variable \(x\) satisfies \(3^{x^{2}} =27\times 9^{x}\), then the value of \(\cfrac{2^{x^{2}}}{\left( 2^{x}\right)^{2}}\) is:

• \(2^{-1}\)

• \(2^{0}\)

• \(2^{3}\)

• \(2^{15}\)

NoteAnswer

• \(2^{-1}\)

• \(2^{0}\)

• \(2^{3}\)

• \(2^{15}\)

NoteSolution

First observe that \(\cfrac{2^{x^{2}}}{\left( 2^{x}\right)^{2}} =2^{x^{2} -2x}\). Now taking \(\log_{3}\) on both sides of the given equation and using the properties of logarithms:

\[ \begin{array}{ r r l } & 3^{x^{2}} & =27\times 9^{x}\\ \Longrightarrow & x^{2}\log 3 & =\log 27+x\log 9\\ \Longrightarrow & x^{2} & =3+2x\\ \Longrightarrow & x^{2} -2x & =3 \end{array} \]

Therefore, the value of the expression is \(2^{3}\).