This worksheet explores the natural logarithm as the inverse function of the exponential function $f(x) = e^x$.

Here's a breakdown of the questions and tasks:

1. Identify Values for $f(x) = e^x$:

   • You are asked to approximate the values of $f(x) = e^x$ for given $x$ values $\{0, 1, \frac{1}{2}, 1, 2, e, 3, 4\}$.
   • Fill these values into a table.

2. Sketch the Inverse Function:

   • You are then asked to sketch the inverse function by swapping $x$- and $y$-values, as this will form the table for $\ln(x)$ as the inverse of $e^x$.
   • This table and graph should reflect the relationship where each $x$-value corresponds to a $y$-value under the function $f(x) = e^x$, and the inverse switches these.

3. Investigate the Inverse Function (Logarithmic Properties):

   • Fill in specific values to understand how logarithms operate as inverses.
   • This section shows that $\ln(x)$ provides the exponent $x$ such that $e^x = y$, where $y$ is the result of $f(x) = e^x$.

To help, here are some key points:

• The Inverse Relationship: $f(x) = e^x$ and $f^{-1}(x) = \ln(x)$ mean that $e^{\ln(x)} = x$.
• Key Values: The given values (like $\ln(1) = 0$, $\ln(e) = 1$, and $\ln(7.389) \approx 2$) represent exact points that describe this relationship.
• Graphing: Ensure the exponential curve and its inverse, the logarithmic curve, are plotted accurately to reflect the symmetry along $y = x$.

Would you like a step-by-step solution or further clarification on any part?

Here are five related questions that might help expand on this topic:

1. What are the characteristics of the exponential function $f(x) = e^x$ in terms of growth rate and asymptote?
2. How does the logarithmic function behave as the inverse of an exponential function in terms of its domain and range?
3. How would the graph of $f(x) = e^x$ differ if the base were a number other than $e$?
4. Why does $\ln(e^x) = x$ hold true, and how is this property used in simplifying logarithmic expressions?
5. How does the value of $e$ (approximately 2.718) influence natural logarithmic and exponential functions?

Tip: When working with exponentials and logarithms, remember that they are inverses of each other, allowing you to switch between multiplication (in exponential form) and addition (in logarithmic form) easily.