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In mathematics, there are several logarithmic identities. Using simpler operations Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. Cancelling exponentials Logarithms and exponentials (antilogarithms) with the same base cancel each other. This is true because logarithms and exponentials are inverse operations (just like multiplication and division). Changing the base This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log10, but not for log2. To find log2(3), you have to calculate log10(3) / log10(2) (or ln(3)/ln(2), which is the same thing). This formula has several consequences: Logarithmic "snake" identity Any sequence can be simplified to... This leads to the identity Summation/subtraction The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities: Trivial identities is undefined because there is no number such that . limit (mathematics)|Limits The last limit is often summarized as "logarithms grow more slowly than any power or root of x". note: to say the limit of a function "equals infinity" is not strictly correct notation, as "infinity" is not a value. What is meant by the limits equations above is simply that the functions increase/decrease without bound. Derivatives of logarithmic functions Integral definition Integrals of logarithmic functions To remember higher integrals, it's convenient to define: where is the nth harmonic number. So, for example, the first few are: Then, | |||||||
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