Natural Logarithm Function

The exponential function is continuous and strictly increasing from \(\mathbb{R}\) to \((0, +\infty)\). By the bijection theorem, for any strictly positive real number \(x\), there exists a unique real number \(y\) such that \(e^y = x\). This unique solution \(y\) is called the natural logarithm of \(x\) and is denoted by \(\ln(x)\).

The natural logarithm function, denoted by \(\ln\), is the function defined on \((0, +\infty)\) which associates each real number \(x > 0\) with the unique solution of the equation \(e^y = x\) where \(y\) is the unknown. We write:$$ y = \ln(x) $$

For all \(x > 0\) and any real \(y\):

• \(e^y = x \iff y = \ln(x)\)
• \(e^{\ln(x)} = x\)
• \(\ln(e^y) = y\)

Particular values:
\(\ln(1) = 0\) (since \(e^0 = 1\)) and \(\ln(e) = 1\) (since \(e^1 = e\))

The graphs of the exponential and natural logarithm functions are symmetric with respect to the line \(y = x\).

For all strictly positive real numbers \(a\) and \(b\):$$\ln(a \times b) = \ln(a) + \ln(b)$$

For all \(a > 0\), \(b > 0\) and any integer \(n\):

• Inverse: \(\ln\left(\frac{1}{b}\right) = -\ln(b)\)
• Quotient: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• Power: \(\ln(a^n) = n\ln(a)\)
• Square Root: \(\ln(\sqrt{a}) = \frac{1}{2}\ln(a)\)

The function \(\ln\) is differentiable and continuous on \((0, +\infty)\). For all \(x > 0\):$$\ln'(x) = \frac{1}{x}$$Since \(1/x > 0\) on its domain, the natural logarithm is strictly increasing on \(]0, +\infty[\).

The natural logarithm has the following end behaviors:

• \(\displaystyle\lim_{x \to +\infty} \ln(x) = +\infty\)
• \(\displaystyle\lim_{x \to 0^+} \ln(x) = -\infty\)

The line \(x = 0\) (the y-axis) is a vertical asymptote to the curve of \(\ln\).

Let \(u\) be a differentiable and strictly positive function on an interval \(I\). The function \(f(x) = \ln(u(x))\) is differentiable on \(I\) and:$$\textcolor{colorprop}{(\ln(u(x)))' = \frac{u'(x)}{u(x)}}$$