If x satisfies the equation 4^x = 256, then x is equal to _____.
Understand the Problem
The question is asking to solve the equation 4^x = 256 for the variable x. This involves recognizing that both sides of the equation can be expressed as powers of 2 to find the value of x.
Answer
$x = 4$
Answer for screen readers
The value of $x$ is $4$.
Steps to Solve
-
Express Both Sides as Powers of 2
We know that $4$ can be expressed as $2^2$. Therefore, we can rewrite the equation:
$$ 4^x = (2^2)^x = 2^{2x} $$ -
Identify the Power of 2 on the Right Side
Next, we need to express $256$ as a power of $2$. We know that:
$$ 256 = 2^8 $$ -
Set the Exponents Equal to Each Other
Since both sides now have the same base, we can set the exponents equal to each other:
$$ 2x = 8 $$ -
Solve for x
Now we can solve for $x$ by dividing both sides by $2$:
$$ x = \frac{8}{2} = 4 $$
The value of $x$ is $4$.
More Information
The equation $4^x = 256$ demonstrates the properties of exponents. Since both sides can be expressed using the base of 2, it becomes straightforward to solve for $x$.
Tips
- Ignoring Base Conversion: A common mistake is to forget to convert the bases to a common base before equating exponents. Always ensure the bases are the same when manipulating exponent equations.
- Arithmetic Errors: When calculating $8 / 2$, be careful not to miscalculate.
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