Logarithms

Definition: the exponent or power to which a base must be raised to yield a given number. Logarithms are the inverse of exponents.

If $n = \log_a x$ then $a^n = x$, so $\log_a x$ tells what exponent is needed to make x from a.

Laws of Logarithms

• $\log_a {1} = 0$
• $\log_a {a} = 1$
• $\log_a {(x \times y)} = \log_a {x} + \log_a {y}$
• $\log_a {x^y} = y\log_a {x}$
• $\log_a {\frac{1}{y}} = -\log_a {y}$
• $\log_a {\frac{x}{y}} = \log_a {x} - \log_a {y}$
• $\log_b {x} = (\log_b {a}) \times \log_a {x}$

TODO: what does the last law mean?

Base Transformation Law

$$\log_a {x} = \frac{\log_b {x}}{\log_b {a}}$$

Change of base formula