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 $\ln(e^x) = x$:

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

3. Solve for $x$ by dividing both sides by 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.