If x satisfies the equation 4^8x = 256, then x is equal to.

Understand the Problem

The question is asking to solve for x in the equation 4^8x = 256. This involves understanding exponential equations and possibly utilizing logarithms to find the value of x.

Answer

$\frac{1}{2}$

Steps to Solve

Rewrite Equation with the Same Base To solve the equation (4^{8x} = 256), first express both sides using the same base. Note that (4) can be expressed as (2^2) and (256) can be expressed as (2^8):

$$ (2^2)^{8x} = 2^8 $$

Apply the Power of a Power Property Using the exponent rule ((a^m)^n = a^{m \cdot n}), we can simplify the left side:

$$ 2^{16x} = 2^8 $$

Set Exponents Equal to Each Other Since the bases are the same, we can set the exponents equal to one another:

$$ 16x = 8 $$

Solve for x Now, divide both sides by (16) to isolate (x):

$$ x = \frac{8}{16} = \frac{1}{2} $$

The value of (x) is $\frac{1}{2}$.

More Information

In this problem, we utilized properties of exponents to solve for the variable (x) in an exponential equation. Equating the exponents after simplifying allowed us to find the value of (x$ directly.

Tips

Not rewriting both sides of the equation using the same base can complicate the solution. Always ensure bases match or use logarithms if necessary.
Forgetting to simplify fractions correctly when solving for (x).

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