3.3 | Properties of Logarithms

Change-of-Base Formula Let \(a,b,\) and \(x\) be positive real numbers such that \(a \not = 1\) and \(b \not = 1.\) Then \(\log_a x\) can be converted to a different base as follows.

  • Base \(b\): $$\displaystyle \log_a x= \frac{\log_b x}{\log_b a}$$
  • Base \(10\): $$\displaystyle \log_a x= \frac{\log x}{\log a}$$
  • Base \(b\): $$\displaystyle \log_a x = \frac{\ln x}{\ln a}$$
Properties of Logarithms Let \(a\) be a positive number such that \(a \not = 1,\) and let \(n\) be a real number. If \(u\) and \(v\) are positive real numbers, the following properties are true.

  • Product Property: $$\log_a(u \cdot v) = \log_a u + \log_a v$$
  • Quotient Property: $$\log_a\frac{u}{v} = \log_a u – \log_a v$$
  • Power Property: $$\log_a u^n = n \log_a u$$

E 3.3 Exercises