To solve the natural logarithm equation [tex]\(\ln x = 6\)[/tex], we can rewrite it as an exponential equation. Here's the step-by-step method to do that:
1. Understand the relationship between natural logarithms and exponentials: The natural logarithm function, denoted as [tex]\(\ln\)[/tex], is the inverse of the exponential function with base [tex]\(e\)[/tex]. This means that if [tex]\(\ln x = y\)[/tex], it can be written as [tex]\(e^y = x\)[/tex].
2. Apply the exponential form: Given the equation [tex]\(\ln x = 6\)[/tex], we need to express [tex]\(x\)[/tex] in terms of [tex]\(e\)[/tex], the base of natural logarithms. According to the relationship, [tex]\( \ln x = 6 \)[/tex] can be rewritten as:
[tex]\[
x = e^6
\][/tex]
3. Evaluate the exponential expression: Calculating [tex]\(e^6\)[/tex]:
From the relationship [tex]\( \ln x = 6\)[/tex], we find that:
[tex]\[
x = e^6
\][/tex]
Given this, we find that:
[tex]\[
e^6 \approx 403.4287934927351
\][/tex]
So, the solution to [tex]\(\ln x = 6\)[/tex] is:
[tex]\[
x \approx 403.4287934927351
\][/tex]
This means that when [tex]\(x\)[/tex] is approximately 403.4287934927351, the natural logarithm of [tex]\(x\)[/tex] is 6.
