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ContentLogarithms
LogarithmsAn exponential number is a function of the form 𝑏^{𝑛} where 𝑏 is known as the base and 𝑛 is the exponent, power, or index of the exponentiation. In general, logarithms are exponents. If a number, 𝑥, is expressed in form of exponential number, 𝑏^{𝑦}, then the logarithm with base 𝑏 of the number, 𝑥, is equal to 𝑦, the exponent of the exponential number, 𝑥.𝑥=𝑏^{𝑦}⇒
The definition of logarithm is
Definiton (Logarithm)
Logarithm is a function. The logarithm of a numebr, 𝑥, is defined as the power to which a given base, 𝑏, must be raised in order to produce that number. Provided that 𝑥>0, and 𝑏 is any number such that 𝑏>0 and 𝑏≠1.
Systems of LogarithmsA system of logarithms can be produced for a specific base by raising to various powers. The two most common logarithm systems are common logarithms and natural logarithems.Common LogarithmsLogarithms having a base of 10 are called common logarithms. The common logarithm function is usually denoted byNatural LogarithmsLogarithms having a base of ℯ are called natural, hyperbolic, or Napierian logarithms. The natural logarithm function is usually denoted byLaws of LogarithmsLogarithms aid in multiplying, dividing, and raising numbers to higher powers. There are three laws of logarithms, which apply to any base:
Properties of Logarithms
Change of BaseSince most calculators are only capable of evaluating common logarithms and natural logarithms, method of change of base is needed to evaluate any other logarithms other than common logarithms and natural logarithms. The change of base formula is
Proof:
Let 𝑝=
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