To solve for [tex]\( x \)[/tex] in the equation [tex]\( \log_x 9 = 2 \)[/tex], follow these steps:
1. Convert the logarithmic equation to an exponential form:
By definition, if [tex]\( \log_{x}(9) = 2 \)[/tex], this can be rewritten in exponential form as:
[tex]\[
x^2 = 9
\][/tex]
2. Solve the quadratic equation:
The equation [tex]\( x^2 = 9 \)[/tex] can be solved by taking the square root of both sides. This gives us:
[tex]\[
x = \pm 3
\][/tex]
3. Consider the properties of the logarithm base:
The base of a logarithm ([tex]\( x \)[/tex] in this case) must be a positive number greater than 1. Therefore, we discard the negative solution since the base of a logarithm cannot be negative or equal to 1.
4. Select the appropriate value:
This means we only consider the positive solution:
[tex]\[
x = 3
\][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( \log_x 9 = 2 \)[/tex] is:
[tex]\[
\boxed{3}
\][/tex]
