logarithms

A power shows how many times a number (the base) is multiplied by itself, or more generally, how it is raised to an exponent.

If \(a\) and \(b\) are positive real numbers, where \(a \neq 1\), the logarithm of \(b\) to the base \(a\), denoted as \(\log_a(b)\), is defined as the real number \(c\) such that \(a^c = b\).

Exponential equations are equations in which the unknown appears in the exponent of a power. They generally take the form:

\[a^{f(x)} = b^{g(x)}\tag{1}\]

Logarithmic equations are equations in which the unknown appears inside a logarithm. To solve them, it is crucial to understand the properties of logarithms and how these can be applied to isolate and determine the value of the unknown.

Exponential inequalities are inequalities in which the unknown appears in an exponent and are solved using common bases, logarithms, and monotonicity.

The geometric mean belongs to the family of power means. In general form, it is expressed as:
\[
M_g = \left( \prod_{i=1}^{n} x_i \right)^{\frac{1}{n}}
\]