To solve the logarithmic equation \( \log_2(x^4) = 3 \), we can use the properties of logarithms.
First, we can rewrite the equation in its exponential form:
\[ x^4 = 2^3 \]
Calculating \( 2^3 \):
\[ 2^3 = 8 \]
Now we have:
\[ x^4 = 8 \]
To solve for \( x \), we take the fourth root of both sides:
\[ x = 8^{1/4} \]
Calculating \( 8^{1/4} \):
Since \( 8 = 2^3 \), we have:
\[ 8^{1/4} = (2^3)^{1/4} = 2^{3/4} \]
If we want to find a numerical approximation, we can also calculate \( 2^{3/4} \):
Calculating \( 2^{3/4} \) gives us roughly \( 1.681 \).
We also need to note that \( x^4 = 8 \) has one positive solution and one negative solution, which can be determined from the equation:
\[ x = \sqrt[4]{8} \quad \text{or} \quad x = -\sqrt[4]{8} \]
Thus, the possible solutions are \( \sqrt[4]{8} \) and \( -\sqrt[4]{8} \).
Among the potential choices given:
- \( x = 32 \)
- \( x = 24 \)
- \( x = -24 \)
- \( x = -32 \)
None of these match \( x = \sqrt[4]{8} \) or \( x = -\sqrt[4]{8} \).
The solutions we derived do not match any of the provided options. Therefore, none of the given choices is reasonable.
However, if \( x \) must be a valid numerical relation or positive, then only \( x = 32 \) could be considered for evaluation or further computation in similar logarithmic contexts, but it does not solve the given equation.
Overall, the logarithmic equation \( \log_2(x^4) = 3 \) leads to valid solutions not represented in the options. Hence, none of the responses provided are reasonable solutions.
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