Find the value of x under the given condition, ${{9}^{x+2}}=240+{{9}^{x}}$.
  & \left( A \right)0.1 \\
 & \left( B \right)0.2 \\
 & \left( C \right)0.3 \\
 & \left( D \right)0.4 \\
 & \left( E \right)0.5 \\

Answer Verified Verified
Hint: We start solving this question by simplifying the given equation using the exponential formula ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$ and later we use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ to simplify the equation. Then after the simplification we get an equation where bases are equal on both sides. Then we equate the powers on both sides as their bases on both sides are equal. Then by solving the obtained equation we get the required value of x.

Complete step by step answer:
We were given the equation, ${{9}^{x+2}}=240+{{9}^{x}}$.
Now, let us use the formula ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$ to simplify the equation.
Using this formula, we can write ${{9}^{x+2}}$ as,
  & \Rightarrow {{9}^{x+2}}={{9}^{x}}\times {{9}^{2}} \\
 & \Rightarrow {{9}^{x+2}}=81\times {{9}^{x}} \\
So, we can write the given equation as,
  & \Rightarrow {{9}^{x+2}}=240+{{9}^{x}} \\
 & \Rightarrow 81\times {{9}^{x}}=240+{{9}^{x}} \\
 & \Rightarrow 81\times {{9}^{x}}-{{9}^{x}}=240 \\
Now, we can take ${{9}^{x}}$ as common in the above equation, then we can write the above equation as,
  & \Rightarrow \left( 81-1 \right)\times {{9}^{x}}=240 \\
 & \Rightarrow 80\times {{9}^{x}}=240 \\
Now, let us take 80 to the other side. Then we get,
  & \Rightarrow {{9}^{x}}=\dfrac{240}{80} \\
 & \Rightarrow {{9}^{x}}=3...........\left( 1 \right) \\
Now we will write 9 as the square of 3, that is $9={{3}^{2}}$.
Now, let us use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$.
So, we can write ${{9}^{x}}$ as,
  & \Rightarrow {{9}^{x}}={{\left( {{3}^{2}} \right)}^{x}} \\
 & \Rightarrow {{9}^{x}}={{3}^{2x}} \\
So, we can convert the equation (1) as,
  & \Rightarrow {{9}^{x}}=3 \\
 & \Rightarrow {{3}^{2x}}=3 \\
As the bases on the both sides are equal, we can equate the powers on the both sides. So, we get,
  & \Rightarrow 2x=1 \\
 & \Rightarrow x=\dfrac{1}{2} \\
 & \Rightarrow x=0.5 \\
So, the value of x is 0.5.

So, the correct answer is “Option E”.

Note: There is a chance of making a mistake while converting ${{9}^{x+2}}$. One can write it as ${{9}^{x+2}}={{\left( {{9}^{x}} \right)}^{2}}$ and then by writing it as ${{9}^{2x}}$. But it is wrong as the formula we need to use is ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$, so we need to write it as ${{9}^{x+2}}={{9}^{x}}\times {{9}^{2}}$. One can also solve this question by substituting the values of x as given in the options and verifying the equation and eliminating the option if it does not satisfy the equation. But it is a long and hectic way of solving. Suppose taking x=0.1 it is difficult to find the value of ${{9}^{0.1}}$. So, it is better to solve the equation in question than substituting the options and verifying them.