Question

Solve the following equations for x: \[{{2}^{x+1}}={{4}^{x-3}}\].

Answer 1

Hint: In this question, we have to find out the value of x in the question. On the left-hand side, we have two and on the right-hand side, we have four as base. So we have to make the base same on both sides. So we will write four in the form of power of two and then after comparing powers of two on both sides, the value of x will be obtained.

Complete step by step answer:
In the above question, we have to find the value of x in the expression \[{{2}^{x}}^{+1}={{4}^{x-3}}\].
As we know that four is the square of two. So we will write four in the form of power of two which is as follows.
\[\begin{align}
  & {{2}^{x}}^{+1}={{4}^{x-3}} \\
 & \Rightarrow {{2}^{x+1}}={{({{2}^{2}})}^{(x-3)}} \\
\end{align}\]
Now we will multiply \[2\] in \[x-3\] which is in the power of two.
\[{{2}^{x+1}}={{2}^{2x-6}}\]
The bases are the same on the left-hand side and right-hand side. So their powers will also be the same. Hence after comparing the powers of the equations on both sides, we get the following results.
\[x+1=2x-6\]
On taking the numbers with variables on one side and the constant numbers on the other side. We get the following results.
\[\begin{align}
  & 2x-x=6+1 \\
 & \Rightarrow x=7 \\
\end{align}\]
So the value of x is \[7\].

Note:
 If any number has power equals one, then the result will be the same number and if any number has its power equals zero, then the result will be one. If we have a number with negative power then after doing its reciprocal, the power of the number becomes positive.