# Logarithmic Properties

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 $$b$$:
$$\log_a x = \frac{\log_b x}{\log_b a}$$
• Base $$10$$:
$$\log_a x = \frac{\log x}{\log a}$$
• Base $$e$$:
$$\log_a x = \frac{\ln x}{\ln a}$$
Properties of Logarithms:
Let $$a$$ be a positive number $$(a \not = 1)$$, $$n$$ be a real number, and $$u$$ and $$v$$ be positive numbers.

• Product Property:
\begin{aligned} \log_a(uv) & = \log_a u + \log_a v \\ \ln(uv) & = \ln u + \ln v \end{aligned}
• Quotient Property:
\begin{aligned} \log_a \left( \frac{u}{v} \right) & = \log_a u – \log_a v\\ \ln \left( \frac{u}{v} \right) & = \ln u – \ln v \end{aligned}
• Power Property:
\begin{aligned} \log_a u^n &= n \log_a u \\ \ln u^n &= n \ln u \end{aligned}
• $$\log_a a = 1$$
• $$\log_a 1 = 0$$