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docs/mathematics/calculus/transcendental-functions/exponential-and-logarithmic-functions.md

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+# Exponential and logarithmic functions
+
+## The natural logarithm 
+
+The natural logarithm is defined as having its derivative equal to $\frac{1}{x}$. For $x > 0$, then
+
+$$
+\frac{d}{dx} \ln x = \frac{1}{x}.
+$$
+
+### Standard limit
+
+$$
+\lim_{h \to 0} \frac{\ln (1+h)}{h} = 1
+$$
+
+## The exponential function
+
+The exponential function is defined as the inverse of the natural logarithm
+
+$$
+\ln e^x = x.
+$$
+
+Furthermore $e$ may be defined by,
+
+$$
+\begin{array}{ll}
+\lim_{n \to \infty} (1 + \frac{1}{n})^n = e, \\
+\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x.
+\end{array}
+$$
+
+### Derivative of exponential function
+
+The derivative of $y = e^x$ may be calculated by [implicit differentation](../differentation.md#implicit-differentation):
+
+$$
+\begin{array}{ll}
+y = e^x &\implies x = \ln y, \\
+        &\implies 1 = \frac{1}{y} \frac{dy}{dx}, \\
+        &\implies \frac{dy}{dx} = y = e^x.
+\end{array}
+$$
+
+### Standard limit
+
+$$
+\lim_{h \to 0} \frac{e^h - 1}{h} = 1
+$$