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0.1: Logs and Exponents

  • Page ID
    7446
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    You probably learned (and then forgot) these identities in middle school or high school: \[\begin{aligned} \left(x^{a}\right)\left(x^{b}\right) &=x^{a+b} \\ \left(x^{a}\right)^{b} &=x^{a b} \\ \log _{x}(a b) &=\log _{x} a+\log _{x} b \\ a \log _{x} b &=\log _{x}\left(b^{a}\right) \end{aligned}\] Well, it’s time to get reacquainted with them again.

    In particular, never ever write \(\left(x^{a}\right)\left(x^{b}\right)=x^{a b}\). If you write this, your cryptography instructor will realize that life is too short, immediately resign from teaching, and join a traveling circus. But not before changing your grade in the course to a zero.


    This page titled 0.1: Logs and Exponents is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mike Rosulek (Open Oregon State) .

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