Answer :

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]