Common logarithm

Graph of common logarithm

APN2002 Table 1, 1000-1500.agr.f

Slide rule example2

Logarithm keys

Common logarithm refers to the logarithm with base 10. It is denoted as log₁₀(x) or sometimes simply as log(x), especially in contexts where base 10 is assumed by default. The common logarithm of a number is the power to which 10 must be raised to obtain that number. For example, the common logarithm of 100 is 2, because 10 raised to the power of 2 is 100.

Definition

The common logarithm of a positive real number x is the exponent by which 10 must be raised to yield x. Mathematically, this is expressed as:

log₁₀(x) = y ⇔ 10^y = x

For x > 0, the common logarithm log₁₀(x) is defined, and it is a real number. The function is undefined for non-positive values of x.

Properties

The common logarithm has several important properties that make it useful in various fields of mathematics and science. These include:

• Logarithm of 1: log₁₀(1) = 0, because 10⁰ = 1.
• Logarithm of 10: log₁₀(10) = 1, as 10¹ = 10.
• Product Rule: log₁₀(xy) = log₁₀(x) + log₁₀(y).
• Quotient Rule: log₁₀(x/y) = log₁₀(x) - log₁₀(y).
• Power Rule: log₁₀(xⁿ) = n log₁₀(x).

These properties are derived from the fundamental properties of logarithms.

Applications

The common logarithm is widely used in science, engineering, and mathematics for several reasons:

• Simplifying calculations: Before the advent of digital calculators, common logarithms were used to simplify complex multiplications, divisions, and exponentiations into simpler addition and subtraction operations using logarithm tables or slide rules.
• Decibel scale: The decibel scale, used to measure sound intensity and other ratios in a logarithmic fashion, is based on the common logarithm.
• pH scale: The pH scale, which measures the acidity or basicity of an aqueous solution, is defined using the negative of the common logarithm of the hydrogen ion concentration.
• Richter scale: The Richter scale, used to quantify the energy released by earthquakes, is a logarithmic scale based on the common logarithm.

Historical Note

The concept of logarithms, including the common logarithm, was introduced in the early 17th century by John Napier and independently by Joost Bürgi. Logarithms greatly facilitated calculations by transforming multiplicative processes into additive ones. The common logarithm, with its base of 10, was particularly useful for computations in the decimal number system and became widely adopted in scientific and engineering disciplines.

See Also

• Logarithm
• Natural logarithm
• Exponential function
• Decibel
• pH
• Richter scale