To solve for [tex]\( x \)[/tex] in the equation [tex]\(\log_6 x = -2\)[/tex], we need to convert the logarithmic equation to its exponential form.
1. Start with the given equation:
[tex]\[
\log_6 x = -2
\][/tex]
2. Convert the logarithmic equation to exponential form:
Recall that [tex]\(\log_b a = c\)[/tex] means [tex]\(b^c = a\)[/tex]. Here, our base [tex]\(b\)[/tex] is 6, and our exponent [tex]\(c\)[/tex] is -2. So, we can write:
[tex]\[
6^{-2} = x
\][/tex]
3. Simplify the expression [tex]\(6^{-2}\)[/tex]:
When you have a negative exponent, you can rewrite it as a positive exponent in the denominator:
[tex]\[
6^{-2} = \frac{1}{6^2}
\][/tex]
4. Calculate [tex]\(6^2\)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
5. Substitute back into the fraction:
[tex]\[
6^{-2} = \frac{1}{36}
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = \frac{1}{36}
\][/tex]
To express this in decimal form, we can further simplify:
[tex]\[
\frac{1}{36} \approx 0.02778
\][/tex]
So, the simplified answer is:
[tex]\[
x \approx 0.02778
\][/tex]
Thus:
[tex]\[
x = 0.027777777777777776
\][/tex]
