A deep understanding of the exponential function $e^x$ and the natural logarithm $\ln(x)$ is essential. These functions are not only foundational in calculus but also have significant applications in various scientific and mathematical contexts.

**Definitions and Properties**

**Exponential Function **$e^x$

The exponential function, denoted as $e^x$, is defined for all real numbers $x$. It represents the constant $e$ (approximately 2.71828) raised to the power of $x$.

**Natural Logarithm **$\ln(x)$

The natural logarithm, denoted as $\ln(x)$, is the inverse function of the exponential function $e^x$. It is defined for all positive real numbers $x$ and represents the power to which $e$ must be raised to obtain $x$.

**Graphical Representations**

**Graph of **$e^x$

- The graph of $e^x$ is a continuously increasing curve.
- It never touches the x-axis, asymptotic to it, indicating that $e^x$ is always positive.
- The curve passes through the point (0,1), as $e^0 = 1$.

**Graph of **$\ln(x)$

- The graph of $\ln(x)$ is a curve that increases slowly and is undefined for non-positive values of $x$.
- It passes through the point (1,0), since $\ln(1) = 0$.

**Application Examples**

**Example 1: **

Graph $y = e^{2x}$ and $y = e^{-x}$.

**Solution:**

**Summary of their characteristics:**

- $y = e^{2x}$
**:**This is an exponential growth function. The graph is a steeply increasing curve, reflecting the rapid increase of $e^{2x}$ as $x$ becomes larger. The function grows faster than $e^x$ due to the doubling effect of the exponent. - $y = e^{-x}$
**:**This is an exponential decay function. The graph is a decreasing curve, approaching the x-axis as $x$ increases, but never actually touching the x-axis. This reflects the property of exponential decay, where the function values become increasingly small as $x$increases, but never reach zero.

**Example 2: **

Solve $e^{2x} = 7$.

**Solution:**

1. Apply the natural logarithm to both sides of the equation to utilize the property that $\ln(e^x) = x$:

$\ln(e^{2x}) = \ln(7)$2. Simplify the left side by using the property of logarithms that $\ln(e^x) = x$:

$2x = \ln(7)$3. Solve for $x$ by dividing both sides by 2:

$x = \frac{\ln(7)}{2}$The precise solution for $x$ is approximately 0.9729550745276566.

The graph shows the function $e^{2x}$ along with the line $y = 7$. The point where the curve intersects the line $y = 7$ represents the solution to the equation, which corresponds to the x-value we calculated.