Determine which of the following is a reasonable solution to the logarithmic equation log2(x4)=3

Question

log
2
⁡
(
𝑥
4
)
=
3
.(1 point)
Responses

x=32
𝑥
=
32
x equals 32

x=−32
𝑥
=
−
32
x equals negative 32

x=24
𝑥
=
24
x equals 24

x=−24

Answer 1

To solve the logarithmic equation \( \log_2(x^4) = 3 \), we can first rewrite the equation in its exponential form.

Recall that if \( \log_b(a) = c \), then \( a = b^c \). Here, we have:

\[ x^4 = 2^3 \]

Calculating \( 2^3 \):

\[ x^4 = 8 \]

Next, we find \( x \) by taking the fourth root of both sides:

\[ x = \sqrt[4]{8} \]

We can express 8 as \( 2^3 \):

\[ x = \sqrt[4]{2^3} = 2^{3/4} \]

Calculating \( 2^{3/4} \) yields a positive value. The approximate value for \( 2^{3/4} \) is about 1.68 (since \( 2^3 = 8 \), and thus \( \sqrt[4]{8} \) is positive).

Now, let’s check each of the given responses:

1. \( x = 32 \):
   \( \log_2(32^4) = \log_2(2^{30}) = 30 \) (not equal to 3).

2. \( x = -32 \):
   Logarithm of a negative number is undefined.

3. \( x = 24 \):
   \( \log_2(24^4) \) is unlikely to simplify to 3 upon further checks.

4. \( x = -24 \):
   Logarithm of a negative number is undefined.

None of the options provided gives the appropriate logarithmic equation equal to 3. However, the solution \( x = 2^{3/4} \) is the only valid and reasonable solution, which does not appear among your options.

Thus, it seems the question might be formulated incorrectly, but based on logarithmic rules, only non-negative x is valid here.

From the given choices, none are valid solutions to the equation \( \log_2(x^4) = 3\).