# 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: $$\log_a u^n = n \log_a u$$