Natural Exponential Function

Definition

The natural exponential function, denoted as $e^x$, is a fundamental function in mathematics that describes continuous exponential growth or decay. It is the base-$e$ exponential function, where $e$ is the mathematical constant approximately equal to 2.718. The natural exponential function is a crucial concept in the study of exponential functions and their graphs.

5 Must Know Facts For Your Next Test

1. The natural exponential function, $e^x$, is a special case of the exponential function where the base is the constant $e$.
2. The graph of the natural exponential function is always increasing, with the function value approaching positive infinity as $x$ approaches positive infinity.
3. The natural exponential function is the only exponential function that is its own derivative, meaning that $\frac{d}{dx}e^x = e^x$.
4. The natural exponential function is widely used in various fields, including physics, chemistry, biology, and finance, to model continuous growth and decay processes.
5. The natural logarithm, denoted as $\ln(x)$, is the inverse function of the natural exponential function, meaning that $\ln(e^x) = x$ and $e^{\ln(x)} = x$.

Review Questions

• Explain how the natural exponential function, $e^x$, is related to the general exponential function, $a^x$.
  • The natural exponential function, $e^x$, is a special case of the general exponential function, $a^x$, where the base $a$ is the mathematical constant $e$ (approximately 2.718). The natural exponential function is a fundamental function in mathematics and is widely used to model continuous growth and decay processes in various fields. It is the only exponential function that is its own derivative, meaning that $\frac{d}{dx}e^x = e^x$, which makes it a particularly important and useful function in calculus and other areas of mathematics.
• Describe the key features of the graph of the natural exponential function, $e^x$.
  • The graph of the natural exponential function, $e^x$, has several distinctive features. First, the graph is always increasing, meaning that as the input $x$ increases, the function value $e^x$ also increases. Second, the graph passes through the point $(0, 1)$, as $e^0 = 1$. Third, the graph approaches the $x$-axis asymptotically as $x$ approaches negative infinity, and it approaches positive infinity as $x$ approaches positive infinity. This shape and behavior of the natural exponential function graph are crucial for understanding its applications in modeling continuous growth and decay processes.
• Explain the relationship between the natural exponential function, $e^x$, and the natural logarithm, $\ln(x)$.
  • The natural exponential function, $e^x$, and the natural logarithm, $\ln(x)$, are inverse functions of each other. This means that $\ln(e^x) = x$ and $e^{\ln(x)} = x$. The natural logarithm is the function that undoes the effect of the natural exponential function, and vice versa. This inverse relationship is fundamental in calculus and other areas of mathematics, as it allows for the transformation of exponential and logarithmic expressions, which is essential for solving a wide range of problems involving continuous growth and decay processes.

Related terms

Exponential Function:

A function of the form $f(x) = a^x$, where $a$ is a positive constant called the base of the function.

Euler's Number ($e$): The mathematical constant, approximately equal to 2.718, that is the base of the natural logarithm and the natural exponential function.

Continuous Growth/Decay: A process where the rate of change is proportional to the current value, leading to exponential growth or decay over time.