50 lines
No EOL
959 B
Markdown
Executable file
50 lines
No EOL
959 B
Markdown
Executable file
# Exponential and logarithmic functions
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## The natural logarithm
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The natural logarithm is defined as having its derivative equal to $\frac{1}{x}$. For $x > 0$, then
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$$
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\frac{d}{dx} \ln x = \frac{1}{x}.
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$$
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### Standard limit
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$$
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\lim_{h \to 0} \frac{\ln (1+h)}{h} = 1
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$$
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## The exponential function
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The exponential function is defined as the inverse of the natural logarithm
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$$
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\ln e^x = x.
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$$
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Furthermore $e$ may be defined by,
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$$
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\begin{array}{ll}
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\lim_{n \to \infty} (1 + \frac{1}{n})^n = e, \\
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\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x.
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\end{array}
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$$
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### Derivative of exponential function
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The derivative of $y = e^x$ may be calculated by [implicit differentation](../differentation.md#implicit-differentation):
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$$
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\begin{array}{ll}
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y = e^x &\implies x = \ln y, \\
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&\implies 1 = \frac{1}{y} \frac{dy}{dx}, \\
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&\implies \frac{dy}{dx} = y = e^x.
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\end{array}
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$$
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### Standard limit
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$$
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\lim_{h \to 0} \frac{e^h - 1}{h} = 1
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$$ |