Question

If we have the relation b/w and y as \[x={{2}^{y}}\] and \[\dfrac{9\times {{3}^{2x}}-{{3}^{x}}\times {{3}^{x-2}}}{2}=360,\] find the value of \[y\].

Answer 1

Hint: So here, In the question, we have given that \[x={{2}^{y}}\] and we have to find the value of y for that we have to make the base same for power terms so that we can solve the powers using the indices when the base is same then we can compare the powers that’s why we can solve this question and also you must know the formula we are using in this question the formula is \[\dfrac{{{a}^{n}}\times {{a}^{m}}}{{{a}^{p}}}={{a}^{m+n-p}}\].

Complete step-by-step solution:
So here we have Given that \[x={{2}^{y}}\]
And also, we have the equation given in the question
\[\Rightarrow \dfrac{9\times {{3}^{2x}}-{{3}^{x}}\times {{3}^{x-2}}}{2}=360\]
In the above equation we have to Change the base of power terms into 3 then we will get,
\[\Rightarrow \dfrac{{{3}^{2}}\times {{3}^{2x}}-{{3}^{x}}\times {{3}^{x-2}}}{2}=360\]
And also, we know that When the base is the same then we can add powers. So here we are using the formula for that
 \[\dfrac{{{a}^{n}}\times {{a}^{m}}}{{{a}^{p}}}={{a}^{m+n-p}}\] apply this and then we will get.
\[\Rightarrow \dfrac{{{3}^{2x+2}}-{{3}^{x-2+x}}}{2}=360\]
After reducing the power of the equations then we will get,
\[\Rightarrow {{3}^{2x+2}}-{{3}^{2x-2}}=720\]
We will Take \[{{3}^{2x}}\] common from the above equation then we will get
\[\begin{align}
  & \Rightarrow {{3}^{2x}}({{3}^{2}}-{{3}^{-2}})=720 \\
 & \Rightarrow {{3}^{2x}}(9-\dfrac{1}{9})=720 \\
 & \Rightarrow {{3}^{2x}}\times \dfrac{80}{9}=720 \\
 & \Rightarrow {{3}^{2x}}=81 \\
 & \Rightarrow {{3}^{2x}}={{3}^{4}} \\
\end{align}\]
When the Base Is same so we can compare the powers and then we will solve the above equation then we will get.
\[\begin{align}
  & \Rightarrow 2x=4 \\
 & \Rightarrow x=2 \\
\end{align}\]
We have Given that \[x={{2}^{y}}\] we will put the value of x in the given equation to get the value of y
\[\begin{align}
  & x={{2}^{y}}={{2}^{1}} \\
 & y=1 \\
\end{align}\]
So here we have the value of y = 1
Hence the value of y is 1.

Note: Here in this question we have to take care of the indices rules \[\dfrac{{{a}^{n}}\times {{a}^{m}}}{{{a}^{p}}}={{a}^{m+n-p}}\] we must know the rules if we don’t know then it is very difficult to solve the problem most of the students are done wrong while adding the powers and we must know that when the base is same then we can add the powers of them Another way to solve the problem is assuming \[t={{3}^{x}}\] and then make the expression in terms of t. and then resubstitute the value Then solve this type of problems.