Physics Study Guide/Logs

Review of logs

Been a while since you used logs? Here is a quick refresher for you.

The log (short for logarithm) of a number N is the exponent used to raise a certain "base" number B to get N. In short, ${\log }_{B}N=x$ means that $B^x=N$.

Typically, logs use base 10. An increase of "1" in a base 10 log is equivalent to an increase by a power of 10 in normal notation. In logs, "3" is 100 times the size of "1". If the log is written without an explicit base, 10 is (usually) implied.

$y=10^x\mathrm{o}\mathrm{r}{\log }_{10}y=x$

therefore: log(10–12) = –12

also: log(1000) = 3

Another common base for logs is the trancendental number $e$, which is approximately 2.7182818.... Since $\frac{d}{dx}{e}^x=e^x$, these can be more convenient than ${\log }_{10}$. Often, the notation $\ln x$ is used instead of ${\log }_{e}x$.

The following properties of logs are true regardless of whether the base is 10, $e$, or some other number.

logA + logB = log(AB) logA – logB = log(A/B) log(AB) = B log(A)

Adding the log of A to the log of B will give the same result as taking the log of the product A times B.

Subtracting the log of B from the log of A will give the same result as taking the log of the quotient A divided by B.

The log of (A to the Bth power) is equal to the product (B times the log of A).

A few examples:
log(2) + log(3) = log(6)
log(30) – log(2) = log(15)
log(8) = log(2³) = 3log(2)