Certainly! Let's solve for [tex]\( x \)[/tex] in the equation [tex]\( \log_9(x) = -2 \)[/tex].
### Step-by-Step Solution:
1. Understand the Logarithmic Equation:
The given equation is [tex]\( \log_9(x) = -2 \)[/tex]. This is a logarithmic equation where the base of the logarithm is 9 and the result is [tex]\(-2\)[/tex].
2. Rewrite the Equation Using Exponential Form:
By the definition of logarithms, if [tex]\( \log_b(a) = c \)[/tex], then [tex]\( a = b^c \)[/tex].
In our problem:
[tex]\[
\log_9(x) = -2
\][/tex]
This implies:
[tex]\[
x = 9^{-2}
\][/tex]
3. Evaluate the Exponential Expression:
Now, evaluate what [tex]\( 9^{-2} \)[/tex] means.
[tex]\[
9^{-2} = \frac{1}{9^2}
\][/tex]
[tex]\[
9^2 = 81
\][/tex]
So:
[tex]\[
9^{-2} = \frac{1}{81}
\][/tex]
4. Write the Final Answer:
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = \frac{1}{81}
\][/tex]
When we convert the fraction [tex]\( \frac{1}{81} \)[/tex] to its decimal form, it results in approximately:
[tex]\[
x \approx 0.012345679012345678
\][/tex]
So, the simplified answer to the given logarithmic equation [tex]\( \log_9(x) = -2 \)[/tex] is:
[tex]\[
x = \frac{1}{81}
\][/tex]
And in decimal form, it is approximately:
[tex]\[
x \approx 0.012345679012345678
\][/tex]
