Logarithms are required in various scientific, technological, and mathematical fields. Although the principle underlying their formal formulation dates back to ancient times, the Scottish mathematician John Napier first published the theory in 1614. After several years, Henry Briggs simplified and reformulated the theory. Finally, in 1700, the Swiss mathematician Euler defined the modern theory of logarithms that we all use and know today.


Definition

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 \(a^c = b\).

\[\log_a{b} = c \quad {\text{iff}} \quad a^c = b\]

In simpler terms, the logarithm of a number is the exponent to which a given base must be raised to obtain that number. This is why logarithm is the inverse operation of exponentiation.

  • \(a\) is the base of the logarithm.
  • \(b\) is the argument.

To clarify the concept, let’s consider a classic example:

\[\log{_2}8 = 3 \quad\quad 2^3 =8\]

To understand logarithms, we must review our knowledge of powers because the two concepts are closely related. The following identities directly result from the mathematical principle that logarithm constitutes the inverse operation of exponentiation:

  • \(a^0 = 1 \quad \text{means} \quad \log{_a}1 = 0 \)
  • \(a^1 = a \quad \text{means} \quad \log{_a}a = 1 \)

Since the exponential is always positive, it is not possible to determine the logarithm of a negative number. In formal terms, \(\nexists\) a real number \(c\) belonging to \(\mathbb{R}\) such that \(a^c < 0\).

  • Logarithms with the base of the number \(e\), known as natural or Napierian logarithms, are typically denoted as \( \log a \) without specifying the base.

  • Logarithms with the base of the number \(10\), known as Briggs logarithms, are typically denoted as \( \text{Log} a \) without specifying the base.


Logarithmic Function

The logarithmic function is the inverse of the exponential function. Therefore, its domain and range are inverted compared to the exponential function. A logarithmic function is defined as a function of the form:

\[ y = \log{_a}x \quad \text{with} \quad a \gt 0 \quad a \neq 1 \quad \forall x \in \mathbb{R}^+\]

  • Domain: \(x \in \mathbb{R}^+ \)
  • Range: \(\mathbb{R}\)

Properties

Logarithms have properties that make them useful for manipulating expressions and equations. These properties are closely related to those of exponentials.

  • The product rule states that the logarithm of a product of two numbers is equal to the sum of their logarithms in the same base: \[\log_a(xy) = \log_ax + \log_ay \]

  • The quotient rule states that the logarithm of a quotient of two numbers is equal to the difference of the numerator and the denominator: \[ \log_a{\frac{x}{y}} = \log_ax-\log_ay \] From the previous expression, if the numerator \(x\) is equal to \(1\), we obtain: \[ \log_a{\frac{1}{y}} = -\log_ay \] This means that the logarithm of the reciprocal of a number \(\frac{1}{y}\) is the opposite of its logarithm, and this is called the co-logarithm, indicated as: \[\text{colog}_a{y} = -log_a{y} = \log_a{\frac{1}{y}} \]

  • The property of the logarithm of a power states that the logarithm of a power of a number is equal to the product of the exponent and the logarithm of the base number: \[ \log{_a}x^n = n \cdot \log{_a}x \] This property directly follows from the properties of exponentials, as an expression like \( x^n \) can be understood as the result of multiplying \( x \) by itself \( n \) times.

  • From the previous property and the property of radicals, it follows that the logarithm of a radical is equal to the quotient between the logarithm of the radicand and the index of the root: \[\log_a\sqrt[n]{b} = \frac{1}{n}\log_ab\]


Changing base

It is essential to understand that any logarithm with a certain base \(a\) can be expressed in terms of the ratio between logarithms with different bases. For example, a logarithm with base \(a\) and argument \(b\) can be expressed as the ratio between two logarithms with the same base \(p\), where the argument of the numerator is \(b\) and the argument of the denominator is \(a\):

\[\log{_a}b = \frac{\log{_p}b}{\log{_p}a} \]

This property is helpful since it simplifies calculations significantly in some situations.

Demonstration

Let’s use a simple example to demonstrate the formula for changing the base of the logarithms. According to the definition of logarithm, the logarithm in base \(a\) of a number \(x\), denoted as \(\log_a(x)\), represents the exponent to which we must raise the base \(a\) to get the number \(x\).

We can write the change of base property as: \[\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\]


Let’s consider the substitution \( y = \log_a(x) \), which means, according to the definition of logarithm, \( a^y = x \). We have: \[\log_b(a^y) = \log_b(x)\]


For the property of the power of a logarithm, we obtain: \[y \cdot \log_b(a) = \log_b(x)\]


Now, dividing both sides by \(\log_b(a) \) we obtain: \[y = \frac{\log_b(x)}{\log_b(a)}\]

While \( y = \log_a(x) \), we have proved that:

\[\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\]


Logarithmic Equations

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. A logarithmic equation takes the form:

\[ \log_af(x) = g(x) \]

  • \( a \) is the base of the logarithm and and it must meet the condition \( a \gt 0, a\neq 1 \)

  • \(f(x)\), the argument of the logarithm must be greater than zero. This is because the logarithm function is only defined for positive numbers.