Certainly! Let's tackle the given equation step-by-step. We start with the natural logarithmic equation:
[tex]\[
\ln x = 5
\][/tex]
1. Understanding the Natural Logarithm (ln): The natural logarithm [tex]\(\ln(x)\)[/tex] is the logarithm to the base [tex]\(e\)[/tex], where [tex]\(e\)[/tex] is approximately equal to 2.71828. The natural logarithm function [tex]\(\ln(x)\)[/tex] is the inverse function of the exponential function [tex]\(e^x\)[/tex].
2. Rewriting as an Exponential Equation: To convert the logarithmic form [tex]\(\ln x = 5\)[/tex] to its equivalent exponential form, we use the fact that if [tex]\(\ln x = y\)[/tex], then [tex]\(x = e^y\)[/tex].
3. Applying this conversion to our problem:
[tex]\[
\ln x = 5 \implies x = e^5
\][/tex]
4. Conclusion: Thus, the exponential form of the equation [tex]\(\ln x = 5\)[/tex] is:
[tex]\[
x = e^5
\][/tex]
This exponential form tells us that [tex]\(x\)[/tex] is equal to [tex]\(e\)[/tex] raised to the power of 5, and solving this gives:
[tex]\[
x \approx 148.4131591025766
\][/tex]
Therefore, the solution to the equation [tex]\(\ln x = 5\)[/tex] is [tex]\(x \approx 148.4131591025766\)[/tex].
