Loga, logarithm to base a

Calculator and formula to compute the logarithm on a given base

Logarithm to base a calculator

What is calculated?

The function Loga returns the logarithm x to base a of the given power value y. Both inputs must be positive real numbers and the base must not be 1.

Input values



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

Logarithm to base a info

Properties

Logarithm to base a:

  • Variable base: a > 0, a ≠ 1
  • Domain: (0, ∞)
  • Range: (-∞, ∞)
  • Inverse function of a^x

Note: The base a must be positive and not equal to 1. The power value y must be positive.

Special examples
log₂(8) = 3
2³ = 8
log₃(27) = 3
3³ = 27
log₅(125) = 3
5³ = 125
log₁₆(256) = 2
16² = 256
Restrictions
  • Base a ≠ 1 (otherwise undefined)
  • Base a > 0 (real logarithms)
  • Power value y > 0 (real results)

Formulas of the logarithm to base a

Definition
\[\log_a(y) = x \Leftrightarrow a^x = y\] Basic definition
Change of base
\[\log_a(x) = \frac{\log(x)}{\log(a)} = \frac{\ln(x)}{\ln(a)}\] Convert between bases
Product rule
\[\log_a(x \cdot y) = \log_a(x) + \log_a(y)\] Logarithm of a product
Power rule
\[\log_a(x^n) = n \cdot \log_a(x)\] Logarithm of a power
Quotient rule
\[\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\] Logarithm of a quotient
Base change reciprocal
\[\log_a(x) = \frac{1}{\log_x(a)}\] Reciprocal relationship

Calculation example

Example: calculate log₁₆(256)

Given:

  • Power value y = 256
  • Base a = 16
  • Sought: x = log₁₆(256)

Calculation:

\[\log_{16}(256) = 2\] \[\text{since } 16^2 = 256\]

Interpretation: 2 is the exponent to which base 16 must be raised to obtain 256.

Binary example

Example: Computer science - determine bits

Problem:

How many bits are required to represent 1024 different values?

Solution:

\[\text{Bits} = \log_2(1024) = 10\] \[\text{since } 2^{10} = 1024\]

Application: The base-2 logarithm is commonly used in computer science.

Growth example

Example: exponential growth

Given:

A population grows by factor 3 per time unit. After which time has it grown by factor 81?

Solution:

\[3^t = 81\] \[t = \log_3(81) = 4\] \[\text{since } 3^4 = 81\]

Result: After 4 time units the population grew by factor 81.

Definition and applications

General definition

The logarithm to base a of a number y is the exponent x to which the base a must be raised to obtain y: a^x = y.

\[y = \log_a(x)\]
Practical applications

Logarithms with different bases are used in: computer science (base 2), chemistry (pH value, base 10), exponential functions, half-life calculations, and scaling.

Important properties
  • loga(1) = 0 for all a > 0, a ≠ 1
  • loga(a) = 1 for all a > 0, a ≠ 1
  • loga(a^n) = n for all real n
  • a^(loga(x)) = x for x > 0

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  •