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.

• \(q^{\log_q x}=x, \qquad \log_q \big(q^x\big)=x\)
• \(\log_q x=\frac{\log_p x}{\log_p q}\)
• \(\log_q 1=0, \qquad \log_q q=1\)
• \(\log_q(xy)=\log_q x+\log_q y\)
• \(\log_q\big(\frac{x}{y}\big)=\log_q x-\log_q y\)
• \(\log_q\big(\frac{1}{y}\big)=-\log_q y\text{,}\)
• \(\log_q(x^y)=y\log_q x\)
• \(\lim\limits_{x\rightarrow\infty}\log_q x=\infty, \qquad \lim\limits_{x\rightarrow0}\log_q x=-\infty\)
• 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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