Real Logarithms#

The Real Natural Logarithm#

Theorem: Injectivity of the Real Natural Logarithm

The real exponential function is injective on \((0; \infty)\).

Proof

TODO

Definition: The Real Natural Logarithm

The real natural logarithm is the inverse of the real exponential function on \((0; \infty)\).

Notation

\[\ln(x) \qquad \log_\mathrm{e}(x) \qquad \log(x)\]

Addition and Subtraction Theorem for the Real Natural Logarithm

The real natural logarithm has the following property:

\[\ln (xy) = \ln x + \ln y \qquad \ln \left(\frac{x}{y}\right) = \ln x - \ln y\]

Proof

TODO

Theorem: Continuity of the Real Natural Logarithm

The real natural logarithm is continuous.

Proof

TODO

Theorem: Derivative of the Real Natural Logarithm

The real natural logarithm is differentiable with

\[(\ln x)' = \frac{1}{x}\]

Proof

TODO

Theorem: Antiderivatives of the Real Natural Logarithm

The real natural logarithm is antidifferentiable with

\[\int \ln x = x \ln x - x + C.\]

Proof

We have

\[\ln x = u(x) v'(x)\]

with \(u(x) = \ln x\), \(u'(x) = \frac{1}{x}\), \(v'(x) = 1\) and \(v(x) = x\) for all \(x \in (0, \infty)\). Since \(u\) and \(v\) are continuously differentiable on \((0,\infty)\), we can use integration by parts:

\[\begin{aligned}\int \ln x \,\mathrm{d}x & = u(x) v(x) - \int u'(x) v(x) \,\mathrm{d}x \\ & = x \ln x - \int \frac{1}{x}x\,\mathrm{d}x \\ & = x \ln x - x + C\end{aligned}\]

Theorem: Real Natural Logarithm Variation in Bachmann-Landau

The real natural logarithm \(\ln (x + 1)\) can expressed in Bachmann-Landau notation for \(x \to 0\) as follows:

\[\ln (x+1) = \sum_{k=1}^m \frac{(-1)^{k+1}}{k}x^k + O(x^{m+1}) \qquad \text{for} \qquad x \to 0\]

Proof

TODO

Logarithms#

Definition: Real Logarithm

Let \(b \in \mathbb{R}_{\gt 0}\) and \(b \ne 1\).

The logarithm with base \(b\) is the function \(\log_b: \mathbb{R}_{\gt 0} \to \mathbb{R}\) defined using the real natural logarithm as

\[\log_{b}(x) \overset{\text{def}}{=} \frac{\ln(x)}{\ln(b)}\]

for each \(x \gt 0\).

Notation

We can also write \(\log_b x\).

Theorem: Logarithm as Solution to Equations

If \(b \in \mathbb{R}_{\gt 0}\) and \(b \ne 1\) and \(y \gt 0\), then the equation

\[b^x = y\]

has the only solution

\[x = \log_b y\]

Proof

TODO

Theorem: Multiplication \(\leftrightarrow\) Addition

Logarithms can be used to turn products into sums and viceversa:

\[\log_{b} (a_1 \times \cdots \times a_n) = \log_{b} (a_1) + \cdots + \log_{b} (a_n)\]

Proof

TODO

Theorem: Division \(\leftrightarrow\) Subtraction

Logarithms can be used to turn division into subtraction and vice versa:

\[\log_{b}\left( \frac{a}{c} \right) = \log_{b}(a) - \log_{b}(c)\]

Proof

TODO

Theorem: Exponentiation \(\leftrightarrow\) Multiplication

Logarithms can be used to turn multiplication into exponentiation and vice versa:

\[\log_{b}(a^c) = c\cdot\log_{b}(a)\]

Proof

TODO

Theorem: Base Change

Let \(a \gt 0\) and \(b_1, b_2 \gt 0\) with \(b_1 \ne 1\) and \(b_2 \ne 1\).

We can turn the real logarithm with base \(b_1\) to real logarithms with base \(b_2\) as follows:

\[\log_{b_1} a = \frac{\log_{b_2} a}{\log_{b_2} b_1}\]

Proof

TODO

Theorem: Differentiability of Logarithms

The real logarithm \(\log_b x\) (\(b \gt 0\), \(b \ne 1\)) is differentiable on \((0, +\infty)\) and its derivative can be constructed via the real natural logarithm:

\[(\log_b x)' = \frac{1}{x \ln b}\]

Proof

TODO

Theorem: Antidifferentiability of Logarithms

The real logarithm \(\log_b x\) (\(b \gt 0\), \(b \ne 1\)) is antidifferentiable on \((0, +\infty)\) and its antiderivatives can be constructed via the real natural logarithm:

\[\int \log_b x \,\mathrm{d}x = x \log_b x - \frac{x}{\ln b} + C\]

Proof

TODO