• $\log_{\text{a}}\text{x}$ - logarithm of number $\text{x}$ on the base a (a > 0, a ≠ 1, x > 0)
• $\log \text{x}$ - common logarithms (Logarithms with base 10 are called common logarithms, a = 10).
• ln x - natural logarithms (Logarithms with base e are called natural logarithms, a = e).

NOTE : a, b, x, y are positive real numbers and a ≠ 1, b ≠ 1

• $\log_{\text{a}}(\text{xy})=\log_{\text{a}}\text{x} + \log_{\text{a}}\text{y}$
• $\log_{\text{a}} \left(\dfrac{\text{x}}{\text{y}}\right)=\log_{\text{a}}\text{x} - \log_{\text{a}}\text{y}$
• $\log_{\text{x}}{\text{x}}=1$
• $\log_{\text{a}}{1}=0$
• $\log_{\text{a}}{(\text{x})^\text{m}}=\text{m}(\log_{\text{a}}\text{x})$ where, m is any real number.
• $\log_{\text{a}}{\text{x}}=\dfrac{1}{\log_{\text{x}}{\text{a}}}$
• $\log_{\text{a}}{\text{x}} = \dfrac{\log_{\text{b}}{\text{x}}}{\log_{\text{b}}{\text{a}}} = \dfrac{\log{\text{x}}}{\log{\text{a}}}$
• $\log_\text{a}{0}=\begin{cases}-\infty & \text{if} \ \text{a} > 1\\+\infty & \text{if} \ \text{a} < 1\end{cases}$