Common logarithm

The common logarithm of a number $x$ is the exponent to which we must raise 10 to obtain $x$. In other words, if $10^y = x$, then we say that $y$ is the common logarithm of $x$, and we can write $y = \log_{10}(x)$. We often abbreviate $\log_{10}(x)$ to just $\log(x)$.

Because $10^y$ is always positive, $\log(x)$ is only defined for positive $x$ values. We can plot $\log(x)$ against $x$ to obtain the function graph shown below. This function goes to $-\infty$ as $x$ approaches 0, and it goes off to $\infty$ (very, very slowly) as $x$ tends to $\infty$.

Common Logarithm

The logarithm has a number of properties that follow from its definition as an exponent:

  1. $\log_{10}(x \cdot y) = \log_{10}(x) + \log_{10}(y)$;
  2. $\log_{10}(\frac{1}{y}) = -\log_{10}(y)$;
  3. $\log_{10}(\frac{x}{y}) = \log_{10}(x) - \log_{10}(y)$ (follows from (1) and (2));
  4. $\log_{10}((x)^n) = n \cdot \log_{10}(x)$

Other logarithm functions can be defined for other positive "bases". For example, the other most common logarithm is the natural logarithm: the natural logarithm of $x$, written $\log_{e}(x)$ or $\ln(x)$, is the exponent to which we must raise $e$ to obtain $x$ ($e = 2.7182818284\ldots$).

Any logarithm base will have a function graph having the same general shape as the common logarithm, and any logarithm base will enjoy analogous versions of the properties detailed above.