To solve for [tex]\( x \)[/tex] in the equation [tex]\(\log_{81} x = \frac{1}{2}\)[/tex], we need to convert the logarithmic equation to its exponential form.
Recall the relationship between logarithms and exponents: if [tex]\(\log_b a = c\)[/tex], then this implies [tex]\( b^c = a \)[/tex].
Given:
[tex]\[
\log_{81} x = \frac{1}{2}
\][/tex]
This can be written in exponential form as:
[tex]\[
81^{\frac{1}{2}} = x
\][/tex]
To simplify [tex]\( 81^{\frac{1}{2}} \)[/tex], we recognize that taking the exponent [tex]\(\frac{1}{2}\)[/tex] is the same as finding the square root:
[tex]\[
81^{\frac{1}{2}} = \sqrt{81}
\][/tex]
We know that the square root of 81 is:
[tex]\[
\sqrt{81} = 9
\][/tex]
Thus, we have:
[tex]\[
81^{\frac{1}{2}} = 9
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 9
\][/tex]
