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Logarithmic equations are equations that involve a logarithm with a variable inside.
Understanding how to solve these equations is crucial because they appear in various mathematical contexts.

In the exercise \(\log _{x} 9=\frac{1}{2}\), we have a logarithmic equation where we need to find the base \( x \).
A logarithm \( \log_{b}(a) \) answers the question: To what power must the base \( b \) be raised, to yield the number \( a \)?
Hence, \( \log_{x} 9 = \frac{1}{2} \) tells us the power we need to raise \( x \) to get \( 9 \) is \( \frac{1}{2} \).

Solving logarithmic equations often involves these steps: identifying the equation format, converting it to an exponential form, and solving for the variable.

Converting a logarithmic equation to its exponential form is a vital step in solving it.
The conversion rule is: if \( \log_{b}(a) = c \), then it is equivalent to \[ b^c = a \].

In our specific problem, \( \log_{x} 9 = \frac{1}{2} \) converts to \[ x^{\frac{1}{2}} = 9 \].
This shows the relation between the base and the result via the exponent.

Once in exponential form, sometimes the equation becomes easier to understand and manipulate.
Here, recognizing that raising a number to the power of \( \frac{1}{2} \) is the same as taking the square root is a useful insight before isolating the variable.

After converting to exponential form, the next step is to isolate the variable to solve the equation.
For \[ x^{\frac{1}{2}} = 9 \], we need to get \( x \) by itself.

Since \( x^{\frac{1}{2}} \) is the same as \( \sqrt{x} \), we can square both sides to remove the square root.
Squaring both sides of \[ x^{\frac{1}{2}} = 9 \] results in \[ (x^{\frac{1}{2}})^2 = 9^2 \].

Simplifying, we get: \[ x = 81 \].
This shows the value of \( x \) that satisfies the original logarithmic equation.
The goal of isolating the variable is to express it explicitly to find its value. This might involve algebraic manipulations like squaring both sides, multiplying, or dividing, depending on the form of the equation.