To evaluate [tex]\(\log_b\left(\frac{1}{b^3}\right)\)[/tex], let's start with simplifying the argument of the logarithm:
1. Given the argument [tex]\(\frac{1}{b^3}\)[/tex], we can rewrite it using the property of exponents:
[tex]\[
\frac{1}{b^3} = b^{-3}
\][/tex]
2. Now we have [tex]\(\log_b(b^{-3})\)[/tex].
3. Next, use the fundamental property of logarithms which states that [tex]\(\log_b(b^x) = x\)[/tex]. This property simplifies our expression:
[tex]\[
\log_b(b^{-3}) = -3
\][/tex]
Therefore, the value of [tex]\(\log_b\left(\frac{1}{b^3}\right)\)[/tex] is [tex]\(-3\)[/tex].
So the correct answer is:
[tex]\[
-3
\][/tex]
