# Logarithm (Base a)

< MathGloss‎ | L

This glossary is far from complete. We are constantly adding math terms.
For instructions on adding new terms, please refer to Math Glossary Main Page


Definition
 Logarithm (Base a) The logarithm of $x\,$ to the base $a\,$, denoted by $log_{a}x\,$, is that real number $u\,$ such that $a^u=x\,$ , where $x\gt0\,$ and $a\,$ is a positive constant other than $1\,$. Notice that a logarithm is always an exponent.

## Examples

• $10\,000=10^4\,$. The exponent to which we raise $10\,$ to get $10\,000\,$ is $4\,$, so $\log_{10}10\,000=4\,$

• $8=2^3\,$. The exponent to which we raise $2\,$ to get $8\,$ is $3\,$, so $\log_{2}8=3\,$

• $1=6^0\,$. The exponent to which we raise $6\,$ to get $1\,$ is $0\,$, so $\log_{6}1=0.\,$

• $3=\sqrt{9}=9^{\tfrac {1}{2}}$. The exponent to which we raise $9\,$ to get $3\,$ is $\tfrac {1}{2}\,$, so $\log_{9}3=\tfrac {1}{2}\,$

• $8=8^1\,$. The exponent to which we raise $8\,$ to get $8\,$ is $1\,$, so $\log_{8}8=1\,$