If x satisfies the equation 48^x = 256, then x is equal to _____.

Steps to Solve

1. Apply logarithms to both sides

Take the logarithm of both sides of the equation (48^x = 256). You can use any logarithm base, but common logarithms (base 10) or natural logarithms (base (e)) are typically used.

$$ \log(48^x) = \log(256) $$

2. Use the power rule of logarithms

Utilize the power rule, which states that (\log(a^b) = b \cdot \log(a)).

$$ x \cdot \log(48) = \log(256) $$

3. Isolate x

Now, solve for (x) by dividing both sides by (\log(48)).

$$ x = \frac{\log(256)}{\log(48)} $$

4. Simplify ( \log(256) )

Recognize that (256 = 2^8), so:

$$ \log(256) = 8 \cdot \log(2) $$

5. Simplify ( \log(48) )

Express (48) in terms of its prime factors:

$$ 48 = 16 \cdot 3 = 2^4 \cdot 3 $$

Thus,

$$ \log(48) = \log(2^4) + \log(3) = 4 \cdot \log(2) + \log(3) $$

6. Substitute back to find x

Substituting these into the equation for (x):

$$ x = \frac{8 \cdot \log(2)}{4 \cdot \log(2) + \log(3)} $$

7. Further simplification

This can be somewhat simplified, but for the sake of matching options, we'll evaluate what this means numerically or in relation to the answer choices.