logarithm

  • the logarithm of a positive number: the power of the base that produces the number

  • common logarithm: a logarithm whose base is 10 $\mbox{log} N$

  • natural logarithm: a logarithm whose base is e $\mbox{ln} N$

properties of logarithms (obtained by applying the definition)

我第一次知道properties是这个意思

但我还是不知道property和law的区别

  1. $\mbox{log}_b 1 = 0$

    A logarithm of 11 equals 0.

  2. $\mbox{log}_b b = 1$

    A logarithm of the base equals 1.

  3. $\mbox{log}_b b^n = n$

    A logarithm of a power of the base equals that power

  4. $b ^ {\mbox{log}_b N} = N$

    A base, to a logarithm power with the same base, equals the antilogarithm.

laws of logarithms (used to expand or simplify/condense log experession

  1. $\mbox{log}_b MN = \mbox{log}_b M + \mbox{log}_b N$
    product/sum law
  2. $\mbox{log}_b \frac{M}{N}= \mbox{log}_b {M} - \mbox{log}_b {N}$

    quotient/difference law

  3. $\mbox{log}_b x^n = n\mbox{log}_b x$

    power/coefficient law

  4. $\mbox{log}_b M = \mbox{log}_b N$ if and only if $M=N$

    equal logs law

change-of-base formula

$\mbox{log}_b N = \frac{\mbox{log}_a N} {\mbox{log}_a b}$ where a is the new base

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