Definition of the logarithm function

Given two numbers

${\displaystyle a,b\in {\mathbb{ℝ}}^{+}}$ with ${\displaystyle a\ne 1}$

we define the logarithm of b to base a

${\displaystyle x={\log }_{a}b}$

as the number x to which the real number a (called the base of the logarithm) must be raised to obtain the positive real number b (called the argument of the logarithm). In other words, x is the solution of the equation

${\displaystyle b=a^x}$

Examples of calculating logarithms

For example

• ${\displaystyle {\log }_{2}8=3}$

since ${\displaystyle 2^3=8}$

• ${\displaystyle {\log }_{10}10=1}$

since ${\displaystyle 10^1=10}$

• ${\displaystyle {\log }_{10}100=2}$

since ${\displaystyle 10^2=100}$

• ${\displaystyle {\log }_{a}1=0}$

since ${\displaystyle a^0=1}$ for ${\displaystyle \mathrm{\forall }a\in {\mathbb{ℝ}}^{+}}$

Properties of logarithms

It is easy to show that, given the numbers

${\displaystyle a,b,c\in {\mathbb{ℝ}}^{+}}$ with ${\displaystyle a\ne 1}$

the following properties hold:

${\displaystyle {\log }_a(b\cdot c)={\log }_{a}b+{\log }_{a}c}$

(1)

${\displaystyle {\log }_a\left(\frac{b}{c}\right)={\log }_{a}b-{\log }_{a}c}$

(2)

${\displaystyle {\log }_a\left(b^c\right)=c\cdot {\log }_{a}b}$

(3)

Property (3) actually also holds for values ${\displaystyle c\in \mathbb{ℝ}}$.

Finally, if also ${\displaystyle c\ne 1}$, the change-of-base formula holds

${\displaystyle {\log }_{a}b=\frac{{\log }_{c}b}{{\log }_{c}a}}$

(4)

which makes it possible to compute a logarithm in any base using the base-10 logarithms of a calculator. For example ${\displaystyle {\log }_{2}8=\frac{{\log }_{10}8}{{\log }_{10}2}}$ = 0.903/0.301 = 3.

Why use logarithms?

Logarithms (usually to base 10, and for this reason called decimal logarithms) are extremely useful

• to compress the range of values taken by a quantity (physical or not)

For example, if a quantity has a range that goes from 1 to ${\displaystyle 10^{12}}$ (a million million!), the decimal logarithm of that quantity ranges only from 0 to 12. This makes it possible to represent the quantity in a graph (it is enough to plot on the vertical axis its decimal logarithm instead of the quantity itself).

• Many scales used in the sciences are based on logarithms. Examples are
  • the Richter scale, which measures the magnitude of earthquakes;
  • the scale of the magnitude of stars;
  • the pH scale, which measures the acidity or basicity of substances;
  • the sound intensity scale. If you want to know how it is defined in terms of logarithms, visit the page perception of loudness;
  • the musical scales of pitch are in fact logarithmic scales with respect to the frequency of the fundamental of the corresponding sounds. You can see this, for example, by looking at the keys of the piano: every time you move seven white keys to the right, the frequency of the corresponding note doubles.
• to turn (thanks to properties (1) and (2)) products and quotients into sums and differences

this is particularly useful when dealing with quantities defined through products or ratios of homogeneous quantities. For example, musical intervals are defined through the frequency ratio between the two sounds of the interval.

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