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I search a general rule for calculating the logarithm of a sum or addition. I know that $$\ln{(a+b)}=\ln{\left(a\left(1+\frac b a\right)\right)}=\ln{(a)}+\ln{\left(1+\frac b a\right)}$$ but when the sum implies more terms, how to generalize its calculation/computation? For example when $$\ln{(a+b+c)}$$ or when $$\ln{\left(\sum_i{a_i}\right)}$$ like when we have to compute the denominator of a Bayes formula using log-likelihoods? Thanks for your incoming help.

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1 Answer

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Same thing you can do with more than three terms $$\ln { (a+b+c) } =\ln { \left( \left( a+b \right) \left( 1+\frac { c }{ a+b } \right) \right) } =\ln { \left( a+b \right) +\ln { \left( 1+\frac { c }{ a+b } \right) } } =$$ $$ =\ln { \left( a\left( 1+\frac { b }{ a } \right) \right) +\ln { \left( 1+\frac { c }{ a+b } \right) } } =\ln { a+\ln { \left( 1+\frac { b }{ a } \right) } +\ln { \left( 1+\frac { c }{ a+b } \right) } } $$

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