To solve the expression [tex]\(\ln e^{-9}\)[/tex], we can use properties of logarithms and exponents. Here's a step-by-step solution:
1. Understanding the notation: [tex]\(\ln\)[/tex] refers to the natural logarithm, which is the logarithm to the base [tex]\(e\)[/tex], where [tex]\(e\)[/tex] is approximately 2.71828.
2. Applying the logarithm property: One of the fundamental properties of logarithms is that [tex]\(\ln(e^x) = x\)[/tex]. This property is true because the natural logarithm function and the exponential function are inverses of each other.
3. Using the property on the given expression:
[tex]\[
\ln(e^{-9})
\][/tex]
According to the logarithmic property mentioned above, if we have [tex]\(\ln(e^x)\)[/tex], we can simplify it directly to [tex]\(x\)[/tex].
4. Simplifying the expression:
[tex]\[
\ln(e^{-9}) = -9
\][/tex]
Therefore, the value of [tex]\(\ln e^{-9}\)[/tex] is [tex]\(-9\)[/tex].
