Log, base 10 logarithm

Calculator and formula for the base 10 logarithm

Base-10 logarithm calculator

What is calculated?

The Log function returns the base 10 logarithm of the given number (power value). The argument must be a positive real number.

Input values


Result
The result is shown with the selected number of decimal places

Function graph

Base-10 logarithm graph

Graph of the base-10 logarithm function log(x)

Base-10 logarithm info

Properties

Base-10 logarithm:

  • Base: 10
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • Inverse function of 10^x

Note: The base-10 logarithm is defined only for positive numbers. For complex numbers a separate function is available.

Special values
log(1) = 0
Logarithm of 1 is always 0
log(10) = 1
Logarithm of the base is 1
log(100) = 2
log(10²) = 2
log(0.1) = -1
log(10⁻¹) = -1
Related functions

For complex numbers a separate Log function is available: → Complex base-10 logarithm

Formulas of the base-10 logarithm

Definition
\[\log(x) = \log_{10}(x)\] Logarithm to base 10
Conversion with ln
\[\log(x) = \frac{\ln(x)}{\ln(10)}\] Change of base formula
Product rule
\[\log(x \cdot y) = \log(x) + \log(y)\] Logarithm of a product
Power rule
\[\log(x^a) = a \cdot \log(x)\] Logarithm of a power
Quotient rule
\[\log\left(\frac{x}{y}\right) = \log(x) - \log(y)\] Logarithm of a quotient
Derivative
\[\frac{d}{dx}\log(x) = \frac{1}{x \ln(10)}\] Derivative of the base-10 logarithm

Calculation example

Example: calculate log(100)

Given:

  • x = 100
  • Wanted: log(100)

Calculation:

\[\log(100) = 2\] \[\text{since } 10^2 = 100\]

Interpretation: 2 is the exponent to which base 10 must be raised to obtain 100.

Scientific notation

Example: determine orders of magnitude

Problem:

How many digits does the number 1,000,000 have? Determine the order of magnitude using the base-10 logarithm.

Solution:

\[\log(1{,}000{,}000) = \log(10^6) = 6\]
The number has 7 digits (6+1)

Application: The base-10 logarithm is often used to determine orders of magnitude and in scientific notation.

pH value calculation

Practical example: pH value

Given:

Hydrogen ion concentration [H⁺] = 0.001 mol/L

pH value:

\[\text{pH} = -\log([H^+])\] \[\text{pH} = -\log(0.001) = -\log(10^{-3}) = 3\]

Result: The solution is acidic with pH = 3.

Definition and applications

Historical background

The base-10 logarithm (also called the Briggsian logarithm) was historically the first practical logarithm. It is also known as the common logarithm.

Practical applications

Base-10 logarithms are used in many areas: pH calculations, decibel scale, Richter scale, scientific notation, and wherever orders of magnitude matter.

Mathematical properties
  • Domain: x > 0
  • Range: all real numbers
  • Monotonicity: strictly increasing
  • Continuity: continuous on (0, ∞)

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More Exp, Log and Root functions

Exponential function base 10x  •  Exponential function base ex  •  Exponential function base xy  •  Modular Exponentiation (PowMod)  •  Logarithm to base 10 (Log)  •  Logarithm to base e (Ln)  •  Logarithm to base a (Loga)  •  Square root (sqrt)  •  Cubic root (cbrt)  •  nth root  •  Hypot  •