Question

How do you solve the given equation ${{2}^{x-3}}=32$?

Answer 1

Hint: We start solving the problem by making use of the fact that $32={{2}^{5}}$ in the given equation. We then make use of the law of exponents that if ${{a}^{m}}={{a}^{n}}$, then $m=n$ to proceed through the problem. We then make the necessary calculations in the obtained equation to get the value of x which is the required solution for the given equation.

Complete step by step answer:
According to the problem, we are asked to solve the given equation ${{2}^{x-3}}=32$.
We have given the equation ${{2}^{x-3}}=32$ ---(1).
We know that $32={{2}^{5}}$. Let us use this result in equation (1).
$\Rightarrow {{2}^{x-3}}={{2}^{5}}$ ---(2).
From laws of exponents, we know that if ${{a}^{m}}={{a}^{n}}$, then $m=n$. Let us use this result in equation (2).
$\Rightarrow x-3=5$.
$\Rightarrow x=5+3$.
$\Rightarrow x=8$.
So, we have found the solution for the given equation (i.e., the value of x) ${{2}^{x-3}}=32$ as 8.

$\therefore $ The solution for the given equation (i.e., the value of x) ${{2}^{x-3}}=32$ is 8.

Note: Whenever we get this type of problems, we try to make use of the laws of exponents to get the required solution. We can also solve the given problem as shown below:
We have given the equation ${{2}^{x-3}}=32$ ---(3).
From the laws of exponents, we know that ${{a}^{m-n}}=\dfrac{{{a}^{m}}}{{{a}^{n}}}$, for $m < n$, $32={{2}^{5}}$. Let us use these results in equation (3).
$\Rightarrow \dfrac{{{2}^{x}}}{{{2}^{3}}}={{2}^{5}}$.
$\Rightarrow {{2}^{x}}={{2}^{5}}\times {{2}^{3}}$ ---(4).
From the law of exponents, we know that ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$. Let us use this result in equation (4).
$\Rightarrow {{2}^{x}}={{2}^{5+3}}$.
$\Rightarrow {{2}^{x}}={{2}^{8}}$ ---(5).
From laws of exponents, we know that if ${{a}^{m}}={{a}^{n}}$, then $m=n$. Let us use this result in equation (5).
$\Rightarrow x=8$.