To solve the equation [tex]\( 4^x = \frac{1}{64} \)[/tex], let's follow these detailed steps:
1. Express the right-hand side of the equation in terms of the same base as the left-hand side:
- We know that [tex]\( 4 \)[/tex] can be written as [tex]\( 2^2 \)[/tex].
- We also know that [tex]\( 64 \)[/tex] is a power of 2. Specifically, [tex]\( 64 = 2^6 \)[/tex].
- Therefore, [tex]\( \frac{1}{64} \)[/tex] can be written as [tex]\( \frac{1}{2^6} \)[/tex], which is [tex]\( 2^{-6} \)[/tex].
2. Express both sides of the equation with the same base:
- Recall that [tex]\( 4 \)[/tex] is [tex]\( 2^2 \)[/tex], so [tex]\( 4^x \)[/tex] can be written as [tex]\( (2^2)^x \)[/tex].
- Using the properties of exponents, [tex]\((2^2)^x \)[/tex] simplifies to [tex]\( 2^{2x} \)[/tex].
- Therefore, we can rewrite the equation as [tex]\( 2^{2x} = 2^{-6} \)[/tex].
3. Set the exponents equal to each other:
- Since the bases are now the same, we can equate the exponents.
- This gives us [tex]\( 2x = -6 \)[/tex].
4. Solve for [tex]\( x \)[/tex]:
- Divide both sides of the equation by 2:
[tex]\[
x = \frac{-6}{2}
\][/tex]
- Simplifying this, we get:
[tex]\[
x = -3
\][/tex]
Hence, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{-3}\)[/tex].
