BASIC LOG LAWS

It is impossible to understand the decibel if you do not understand the logarithm. The fundamental idea of the logarithm is based on indices, or the number of times you multiply a number by itself. Indices are often called powers, for example 10 to the power of 3 equals 10 multiplied by itself 3 times (10 * 10 * 10 = 1000). Given any number, for example 1000, a logarithm tells you how many times a particular number must be multplied by itself to give that result. This means that you first have to choose a number, of the base of the logarithm. In acoustics the base 10 is almost always used. Thus:

For example, to use some simple numbers:

Finally, when manipulating logs, regardless of base, the following relationships apply:

log (MN) = log M + log N

log (M/N) = log M - log N

log (1/N) = -log N

log (Nⁿ) = n log N

log (N(m/n)) = (m/n) log N

log _a (M) = log _b (M) * log _a (b)

LOG GRAPHS

It is also important to understand how using a logaritmic scale can affect a graph. A log scale can be used to more adequately display detail within a curve when it contains values over a wide range. For example, the following two graph show roughly the same information, displayed using a linear and logarithmic vertical scale.

Linear vertical scale

Logarithmic vertical scale

Sometimes, a logarithmic scale is also used along the horizontal axis. This is almost always the case when frequency is the unit of the X axis. Note that the distance along the axis between 100Hz and 1Kz is the same as from 1kHz to 10kHz. The meaning of each vertical dotted line therefore changes at each jump, from 10Hz at the start, to 100Hz after the first jump and then to 1000Hz after the second.

Logarithmic horizontal scale