Logarithmic identities

Re­call that \(\log_b(n)\) is defined to be the (pos­si­bly frac­tional) num­ber of times that you have to mul­ti­ply 1 by \(b\) to get \(n.\) Log­a­r­ithm func­tions satisfy the fol­low­ing prop­er­ties, for any base \(b\):

  • In­ver­sion of ex­po­nen­tials: \(b^{\log_b(n)} = \log_b(b^n) = n.\)

  • Log of 1 is 0: \(\log_b(1) = 0\)

  • Log of the base is 1: \(\log_b(b) = 1\)

  • Mul­ti­pli­ca­tion is ad­di­tion in logspace: \(\log_b(x\cdot y) = log_b(x) + \log_b(y).\)

  • Ex­po­nen­ti­a­tion is mul­ti­pli­ca­tion in logspace: \(\log_b(x^n) = n\log_b(x).\)

  • Sym­me­try across log ex­po­nents: \(x^{\log_b(y)} = y^{\log_b(x)}.\)

  • Change of base: \(\log_a(n) = \frac{\log_b(n)}{\log_b(a)}\)

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