Physics Study Guide/Logs

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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, ${\displaystyle \log _{B}\ N=x}$ means that ${\displaystyle B^{x}=N}$.

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.

 ${\displaystyle y=10^{x}\ \mathrm {or} \ \log _{10}y=x\ }$ therefore: log(10–12) = –12 also: log(1000) = 3

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

The following properties of logs are true regardless of whether the base is 10, ${\displaystyle e}$, 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)