The properties of logarithms

The following properties descend directly from the properties of powers and are the basis of every manipulation with logarithms.

Property — Properties of logarithms

For every $a > 0,\ a\ne 1$ and every $x, y > 0$:

\log_a(xy) &= \log_a x + \log_a y & &\text{(product)} \\ \log_a\left(\tfrac{x}{y}\right) &= \log_a x - \log_a y & &\text{(quotient)} \\ \log_a\left(x^k\right) &= k\cdot \log_a x & &\text{(power)} \\ \log_a 1 &= 0,\quad \log_a a = 1 & &\text{(notable values)} \end{aligned}$$

In words: the logarithm of a product is the sum of the logarithms, the logarithm of a quotient is the difference, and an exponent inside the argument “comes down” in front of the logarithm as a factor.

Links

Topics: Logarithmic function
Concepts: Logarithm · Properties of logarithms
Functions: Logarithmic function
Skills: Using formulae