The natural base, denoted as 'e', is an irrational number approximately equal to 2.71828. It's a cornerstone of calculus and appears naturally in many areas of mathematics and science, particularly in growth and decay models.
Common exponential bases are simply numbers raised to a power, like $2^x$, $10^x$, or even $\pi^x$. They're used to model exponential relationships, but unlike 'e', they don't have the same inherent properties that make 'e' so useful in calculus.
| Feature | Natural Base 'e' | Common Exponential Bases (e.g., 2, 10) |
|---|---|---|
| Definition | The limit of $(1 + \frac{1}{n})^n$ as $n$ approaches infinity. | Any number (positive, excluding 1) raised to a power. |
| Derivative | The derivative of $e^x$ is $e^x$ itself. | The derivative of $a^x$ is $a^x \cdot \ln(a)$, where 'a' is the base. |
| Integration | The integral of $e^x$ is $e^x + C$. | The integral of $a^x$ is $\frac{a^x}{\ln(a)} + C$, where 'a' is the base. |
| Natural Occurrence | Appears naturally in continuous growth/decay, compound interest, and probability. | Used for modeling exponential relationships but requires a scaling factor. |
| Logarithm | Its logarithm is the natural logarithm (ln), which simplifies many expressions. | Logarithms to other bases (e.g., base 10) require change-of-base formulas for simplification. |
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