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Mathematical tables: logarithm operations
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Table shows common properties and formulas related to logarithms (logarithm function).

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Logarithms: general formulas#

NameFormula
Logarithm of oneShow sourcelog⁡a1=0\log_{a} 1 = 0
Logarithm of the productShow sourcelog⁡a(x⋅y)=log⁡ax+log⁡ay\log_{a} \left(x \cdot y \right) = \log_{a} x + \log_{a} y
Logarithm of the quotientShow sourcelog⁡axy=log⁡ax−log⁡ay\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y
Sum of logarithms with common baseShow sourcelog⁡ax+log⁡ay=log⁡a(x⋅y)\log_{a} x + \log_{a} y = \log_{a} \left(x \cdot y \right)
Difference of logarithms with common baseShow sourcelog⁡ax−log⁡ay=log⁡axy\log_{a} x - \log_{a} y = \log_{a} \frac{x}{y}
Logarithm from powerShow sourcelog⁡axn=nlog⁡ax\log_{a} x^n = n \log_{a} x
Logarithm from n-th rootShow sourcelog⁡a(xn)=log⁡axn\log_{a} \left( \sqrt[n]{x} \right) = \frac{\log_{a} x}{n}
Logarithm base conversionShow sourcelog⁡ax=log⁡bxlog⁡ba\log_{a} x = \frac{\log_{b} x}{\log_{b} a}

Decimal logarithm #

NameFormula
Decimal logarithm from 1Show sourcelog⁡101=0\log_{10} 1 = 0
Decimal logarithm from 10Show sourcelog⁡1010=1\log_{10} 10 = 1
Decimal logarithm from power of 10Show sourcelog⁡1010n=n\log_{10} 10^n = n
Decimal logarithm to natural conversionShow sourceln⁡x=log⁡10xlog⁡10e≈2.3026⋅log⁡10x\ln x = \frac{\log_{10} x}{\log_{10} e} \approx 2.3026 \cdot \log_{10} x
Natural logarithm to decimal conversionShow sourcelog⁡10x=ln⁡xln⁡10≈0.4343⋅ln⁡x\log_{10} x = \frac{\ln x}{\ln 10} \approx 0.4343 \cdot \ln x

Natural logarithm #

NameFormula
Natural logarithm from 1Show sourceln⁡e=0\ln e = 0
Natural logarithm from e numberShow sourceln⁡e=1\ln e = 1
Natural logarithm from power of e numberShow sourceln⁡en=n\ln e^n = n
Decimal logarithm to natural conversionShow sourceln⁡x=log⁡10xlog⁡10e≈2.3026⋅log⁡10x\ln x = \frac{\log_{10} x}{\log_{10} e} \approx 2.3026 \cdot \log_{10} x
Natural logarithm to decimal conversionShow sourcelog⁡10x=ln⁡xln⁡10≈0.4343⋅ln⁡x\log_{10} x = \frac{\ln x}{\ln 10} \approx 0.4343 \cdot \ln x

Some facts#

  • Calculation of logarithm is an operation opposite to exponentiation.
  • To calculate the logarithm of base a of the number x we ask the question: "to what power we must raise the base to get x".
  • ⓘ Example: The decimal logarithm (of the basis 10) of the number one hundred (100) is two (2), because to obtain this number (100), we would have to raise the base of logarithm (10) to the second power. Formally, we can write this down in the following way:
    log⁡10100=2⇔102=100\log_{10}100 = 2 \Leftrightarrow 10^2 = 100
  • Formally the logarithm is a two arguments operation, where the first argument is the logarithm base, and the second one is the number that we logarithm. In general, we can write:
    log⁡ax=b⇔ab=x\log_{a} x = b \Leftrightarrow a^b = x
    where:
    • a - the logarithm base,
    • x - the number, which we logarithm,
    • b - the result of logarithm operation
  • The logarithm is usually denoted by the symbol log or lg.
  • Sometimes to simplify the expression we omit the base of the logarithm. Then, by default, we assume that the basis is the number ten (10).
  • A special type of logarithm is a natural logarithm. The basis of natural logarithm is so-called number e (sometimes also called the Euler's number):
    e=∑n=0∞1n!=11+11+11⋅2+11⋅2⋅3+⋯e = \sum\limits_{n = 0}^{\infty} \dfrac{1}{n!} = \dfrac{1}{1} + \dfrac{1}{1} + \dfrac{1}{1\cdot 2} + \dfrac{1}{1\cdot 2\cdot 3} + \cdots
  • In general, the logarithm is a inverse function for the power function axa^x. Thus, the natural logarithm is a inverse to the exponential function exe^x.
  • The practical aspect of logarithm is that it allows to replace multiplication with addition, which often simplifies calculations:
    log⁡a(x⋅y)=log⁡a(x)+log⁡a(y)\log_a(x \cdot y) = \log_a(x) + \log_a(y)

  • Logarithm is often used by engineers and in natural science. This is because expressions that have an inherently exponential character (or more generally: power character), become linear after logarithm was applied. For this reason, many laws, e.g. in physics or chemistry have logarithm form.

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