Question

How do you solve ${4^{x - 3}} = 32$?

Answer 1

To solve this problem, we need to know a basic concept which is, if the bases are the same and are in either multiplication or division, then we can add or subtract the exponential numbers. Here in this problem try to make the base term the same with some mathematical manipulations and simplify the terms and atlast find the value of $x$.

Complete step by step solution:
Let us consider the given equation,
${4^{x - 3}} = 32$
Now, we have to make the bases equal so that we can equate the powers. To do that we have to convert the number $32$ in terms of $4$, and the equation becomes,
${4^{x - 3}} = \left( {{4^2}} \right)\left({{4^{\dfrac{1}{2}}}} \right)$
We know that $16 \times 2$ will be $32$, so $16$ can be written as ${4^2}$ and $2$ can be written as ${4^{\dfrac{1}{2}}}$. So I have changed the equation in this way.
Now the bases are equal, so we can equate the powers and simplifying it further we get,
$x - 3 = 2 + \dfrac{1}{2} \\\ x - 3 = \dfrac{5}{2} \\\ x = \dfrac{5}{2} + 3 \\\ x = \dfrac{{11}}{2} \\\$
This is the required solution.

Additional information: Only when the bases of the numbers are in the multiplication or division, we are able to equate the exponential. If the bases are in addition or subtraction, then we should not equate the powers.

Note: In the case, if the number is in the fraction form, then we can reciprocate the denominator and put a negative sign in the exponential number. For instance, let us consider the question $\dfrac{{{4^2}}}{{{4^3}}} = {4^x}$, to reciprocate the number ${4^3}$, we have to put negative sign in the power and it becomes,
${4^2}{4^{-3}}$ = ${4^x}$
$x = 2 - 3 \\\ x = - 1 \\\$
This is the required answer.