Logarithms

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Section A.13 Logarithms

In the following, $x$ and $y$ are arbitrary real numbers that are strictly bigger than 0, and $p$ and $q$ are arbitrary constants that are strictly bigger than one.

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$\displaystyle q^{\log_q x}=x, \qquad \log_q \big(q^x\big)=x$

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$\displaystyle \log_q x=\frac{\log_p x}{\log_p q}$

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$\displaystyle \log_q 1=0, \qquad \log_q q=1$

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$\displaystyle \log_q(xy)=\log_q x+\log_q y$

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$\displaystyle \log_q\big(\frac{x}{y}\big)=\log_q x-\log_q y$

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$\log_q\big(\frac{1}{y}\big)=-\log_q y\text{,}$

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$\displaystyle \log_q(x^y)=y\log_q x$

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$\displaystyle \lim\limits_{x\rightarrow\infty}\log_q x=\infty, \qquad \lim\limits_{x\rightarrow0+}\log_q x=-\infty$

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The graph of $\log_{10} x$ is given below. The graph of $\log_q x\text{,}$ for any $q \gt 1\text{,}$ is similar.
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