To solve the equation [tex]\(\log_4 x = 4\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Understand the logarithmic equation: The equation is in the form of [tex]\(\log_b a = c\)[/tex], where [tex]\(b\)[/tex] is the base of the logarithm, [tex]\(a\)[/tex] is the argument, and [tex]\(c\)[/tex] is the logarithmic value.
2. Convert the log equation to its exponential form: The logarithmic equation [tex]\(\log_4 x = 4\)[/tex] can be rewritten in its exponential form. That is:
[tex]\[
x = 4^4
\][/tex]
3. Calculate the exponential: Compute [tex]\(4^4\)[/tex]. This means raising 4 to the power of 4:
[tex]\[
4^4 = 4 \times 4 \times 4 \times 4
\][/tex]
4. Perform the multiplication step-by-step:
[tex]\[
4 \times 4 = 16
\][/tex]
[tex]\[
16 \times 4 = 64
\][/tex]
[tex]\[
64 \times 4 = 256
\][/tex]
Therefore, [tex]\(4^4\)[/tex] equals 256.
5. Conclude the value of [tex]\(x\)[/tex]: From the above steps, we conclude that:
[tex]\[
x = 256
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\log_4 x = 4\)[/tex] is [tex]\(\boxed{256}\)[/tex].
