To solve the equation [tex]\(\log_4 x = 2 \cdot 5\)[/tex], follow these steps:
1. Simplify the logarithmic expression:
[tex]\[
\log_4 x = 2 \cdot 5
\][/tex]
[tex]\[
\log_4 x = 10
\][/tex]
2. Convert the logarithmic equation to an exponential form:
Recall that [tex]\(\log_b y = z\)[/tex] is equivalent to [tex]\(b^z = y\)[/tex]. So, [tex]\(\log_4 x = 10\)[/tex] can be rewritten as:
[tex]\[
4^{10} = x
\][/tex]
3. Calculate [tex]\(4^{10}\)[/tex]:
[tex]\[
4^{10} = 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4
\][/tex]
When you multiply these out, the value is:
[tex]\[
4^{10} = 1048576
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(1048576\)[/tex], which is not one of the given options (i) [tex]\(12.5\)[/tex], (ii) [tex]\(32\)[/tex], (iii) [tex]\(10\)[/tex], or (iv) [tex]\(20\)[/tex].
Hence, the correct answer is none of the options given.
