Skip to main content

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.

  • \(q^{\log_q x}=x, \qquad \log_q \big(q^x\big)=x\)
  • \(\log_q x=\frac{\log_p x}{\log_p q}\)
  • \(\log_q 1=0, \qquad \log_q q=1\)
  • \(\log_q(xy)=\log_q x+\log_q y\)
  • \(\log_q\big(\frac{x}{y}\big)=\log_q x-\log_q y\)
  • \(\log_q\big(\frac{1}{y}\big)=-\log_q y\text{,}\)
  • \(\log_q(x^y)=y\log_q x\)
  • \(\lim\limits_{x\rightarrow\infty}\log_q x=\infty, \qquad \lim\limits_{x\rightarrow0+}\log_q x=-\infty\)
  • The graph of \(\log_{10} x\) is given below. The graph of \(\log_q x\text{,}\) for any \(q \gt 1\text{,}\) is similar.