Logarithm

Rumman Ansari Software Engineer 2024-07-28 08:58:38 156 Share
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Logarithm Formulas
Definition
Common Logarithm Properties
Product Rule
Quotient Rule
Power Rule
Change of Base Formula
Special Logarithms
Common Logarithm (Base 10)
Natural Logarithm (Base e)
Logarithm of 1
Logarithm of the Base
Logarithm of a Reciprocal
Sum and Difference of Logarithms
Sum
Difference
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Logarithm Formulas

Definition

A logarithm is the power to which a number must be raised to obtain another number. If $b^{y} = x$ , then $\log_{b} (x) = y$ .

\log_{b} (x) = y ⟺ b^{y} = x

Common Logarithm Properties

Product Rule

\log_{b} (x \cdot y) = \log_{b} (x) + \log_{b} (y)

Quotient Rule

\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)

Power Rule

\log_{b} (x^{y}) = y \cdot \log_{b} (x)

Change of Base Formula

\log_{b} (x) = \frac{\log_{k} (x)}{\log_{k} (b)}

Special Logarithms

Common Logarithm (Base 10)

\log_{10} (x) = \log (x)

Natural Logarithm (Base $e$ )

\log_{e} (x) = \ln (x)

Logarithm of 1

\log_{b} (1) = 0 for any base b

Logarithm of the Base

\log_{b} (b) = 1

Logarithm of a Reciprocal

\log_{b} (\frac{1}{x}) = - \log_{b} (x)

Sum and Difference of Logarithms

Sum

\log_{b} (x + y) \neq \log_{b} (x) + \log_{b} (y)

Difference

\log_{b} (x - y) \neq \log_{b} (x) - \log_{b} (y)

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