To solve the given logarithmic equation [tex]\(\log_{36} x = \frac{1}{2}\)[/tex], we need to convert it into its exponential form. A logarithmic equation [tex]\(\log_b a = c\)[/tex] is equivalent to the exponential equation [tex]\(b^c = a\)[/tex].
Given the equation:
[tex]\[
\log_{36} x = \frac{1}{2}
\][/tex]
We can rewrite this in its exponential form:
[tex]\[
36^{\frac{1}{2}} = x
\][/tex]
Next, we simplify the expression [tex]\(36^{\frac{1}{2}}\)[/tex]. The expression [tex]\(36^{\frac{1}{2}}\)[/tex] represents the square root of 36. Therefore,
[tex]\[
36^{\frac{1}{2}} = \sqrt{36}
\][/tex]
We know that the square root of 36 is 6:
[tex]\[
\sqrt{36} = 6
\][/tex]
Thus, we have:
[tex]\[
x = 6
\][/tex]
Therefore, the solution to the equation [tex]\(\log_{36} x = \frac{1}{2}\)[/tex] is:
[tex]\[
x = 6
\][/tex]
