Solve for the Missing Exponent: 4^x = 1/64 Equation

In this problem, we need to determine the exponent of the number on the left side of the equation that makes it equal to the right side.

To do this, our first step is to express the right-hand side as a power of 4. This way, both sides of the equation will have the same base, allowing us to compare the exponents directly.

Let's note that the number 64 is a power of the number 4:

$64=4^3$

Therefore we can write the fraction on the right side like this:

$4^?=\frac{1}{64} \\ 4^?=\frac{1}{4^3}$

Next, we’ll apply the law of negative exponents in reverse:

$\frac{1}{a^n}=a^{-n}$

And we'll express the term on the right side of the equation as a term with a negative exponent with base 4:

$4^?=\frac{1}{4^3} \\ 4^?=4^{-3}$

When we applied the above law of exponents for negative exponents in the opposite direction and expressed the fraction on the right side as a term with a negative exponent,

Now let's examine the equation we got in the last step:

$4^?=4^{-3}$

On both sides of the equation there are terms with identical bases, and therefore we can unambiguously determine that on the left side the exponent must be $-3$ in order for the equation to hold,

Therefore the correct answer is answer B