Question

How do you solve ${{2}^{3x}}=4$?

Answer 1

Hint: In this problem we need to solve the given expression which is an exponential function. We can observe that the variable $x$ is in power of $2$ on the left hand side. On the right hand side we have $4$. So first we will write the number $4$ as ${{2}^{2}}$. Now we can observe that in the obtained equation we have base $2$ in both sides of the equation. Here we have the exponential rule which is ‘when bases are equal, the powers should be equal’. So, we will apply this rule and equate the power of the exponentials in the above equation. Now we will simplify the equation to get the required result.

Complete step by step solution:
Given that, ${{2}^{3x}}=4$.
In the above equation we have exponential having $2$ as base in the left hand side. To solve this equation, we need to convert the right-hand side part into a base of $2$.
On the right hand side we have the number $4$. We can write the number $4$ as ${{2}^{2}}$, then the given equation is modified as
$\Rightarrow {{2}^{3x}}={{2}^{2}}$
In the above equation we can observe that bases of the exponentials on both sides. We have the exponential rule which says that when bases are equal then the powers should be equal. Applying this rule in the above equation, then we will have
$\Rightarrow 3x=2$
Dividing the above equation with $3$ on both sides, then we will get
$\begin{align}
  & \Rightarrow \dfrac{3x}{3}=\dfrac{2}{3} \\
 & \Rightarrow x=\dfrac{2}{3} \\
\end{align}$
Hence the solution of the given equation ${{2}^{3x}}=4$ is $x=\dfrac{2}{3}$.

Note:
We can also check whether the obtained solution is correct or wrong by substituting the calculated value in the given equation. So, substituting $x=\dfrac{2}{3}$ in ${{2}^{3x}}=4$, then we will get
$\begin{align}
  & \Rightarrow {{2}^{3\left( \dfrac{2}{3} \right)}}=4 \\
 & \Rightarrow {{2}^{2}}=4 \\
 & \Rightarrow 4=4 \\
 & \Rightarrow LHS=RHS \\
\end{align}$
Hence the obtained result is correct.