According to the 1797
Britannica a logarithm is:
"...a series of numbers in arithmetical progression, corresponding to others in
geometrical progression; by means of which, arithmetical calculations can be made with much more ease and
expedition than otherwise."
This fundamental property of logarithms makes them utterly indispensible in
engineering and science, where often it is necessary to "mentally" multiply an divide quantities like gain, noise
figure, etc. Thanks to logarithms, those multiplication and division operations are transformed into much simpler
tasks of addition and subtract, respectively.
To make the process even simpler, many of our engineering and science quantities are routinely expressed in
units of decibels. We work with decibels (dB) of gain, decibels wrt a milliwatt of power (dBm), and many other
such quantities. Decibels are commonly used in optics, acoustics, and other realms of physics.
If x = a^{y}, then y = log_{a} x log_{a} (x * y) = log_{a}
x + log_{a} y log_{a} (x / y) = log_{a} x  log_{a} y
log_{a} (x^{n}) = n * log_{a} x 
so,

 log_{10} x is written as log x
 log_{e} x is written as ln x
where "e" is the base of the natural logarithm 
An example would be where you have three stages of gain in series (cascade) that need to be totaled. Supposed
that the first stage quadrupled the power of the input signal (gain = 4), the second stage increased the power by
a factor of 20 (gain = 20), the third stage increased the power by a factor of 2 (gain = 2). In order to calculate
the total power gain, you multiply the gains for a total of 4 x 20 x 2 = 160.
Now, that is not such a hard
mental exercise, but suppose instead the stage gains were as follows: g1 = 7.51 g2 = 22.80 g3 = 3.94
Quick, what is the total gain? If you are a math whiz, you would immediately answer, "674.63832." If you were a
disciplined scientist, however, you would reply, "675," because you would dutifully you know that the answer
cannot be of any greater precision than the lowest precision of any involved quantity. But, I digress.
Now,
let us go about the same exercise using decidels.
G1 = 10 log_{10} (7.51) = 8.76 dB G2 = 10 log_{10} (22.80) = 13.58 dB G3 = 10 log_{10}
(3.94) = 5.95 dB
Total gain is the sum of the three gains = 28.29 dB (ok, 28.3 dB)
To check the
results: 10 log_{10} (674.63832) = 28.3 dB Q.E.D., as they say.
