To solve the problem of converting the logarithmic equation [tex]\(\log_2 x = 24\)[/tex] into an equivalent exponential equation, we need to use the properties of logarithms and exponentials.
The logarithmic equation [tex]\(\log_b a = c\)[/tex] can be rewritten as [tex]\(b^c = a\)[/tex]. Here, [tex]\(b\)[/tex] is the base of the logarithm, [tex]\(a\)[/tex] is the argument (or the value inside the logarithm), and [tex]\(c\)[/tex] is the result of the logarithm.
Applying this property to the given equation:
- The base [tex]\(b\)[/tex] is 2.
- The result [tex]\(c\)[/tex] is 24.
- The argument [tex]\(a\)[/tex] is [tex]\(x\)[/tex].
So, using the property of logarithms:
[tex]\[
b^c = a \implies 2^{24} = x
\][/tex]
This is the equivalent exponential equation. Therefore, the correct answer is:
B. [tex]\(2^{24} = x\)[/tex]
As we calculated, the value of [tex]\(2^{24}\)[/tex] is 16777216, which confirms our understanding and transformation from the logarithmic form to the exponential form.
