- What is a logarithm?
If x = b**y, then y is the logarithm of x for (logarithm) base b. Inverting gives y = log_b(x) where log_b(x) is the logarithm function for base b.

- Note:
- If x = b, then y = log_b(b) = 1.
- If x = 1, then y = 0.
- Note we assumed b≠1 above which is a useless base since with it x can only be 1 and y is indeterminate.

- What is the logarithm of x_1*x_2?
Behold:
log_b(x_1*x_2) = log_(b)[(b**y_1)*(b**y_2)] = log_(b)[b**(y_1+y_2)] = y_1 + y_2 = log_b(x_1)+log_b(x_2)

- What is the logarithm of x**p given
y = log_b(x)?
Behold:
log_b(x**p) = log_b([b**y]**p) = log_b(b**[py]) = py = p*log_b(x) .

- Conversions
between different bases are simple.
Consider
x = b_1**y_1 and x = b_2**y_2 implying b_1**y_1 = b_2**y_2 .

Take the logarithm with respect to general base b to gety_1*log_b(b_1) = y_2*log_b(b_2) or y_2 = [(log_b(b_1)/log_b(b_2)]*y_1 .

If b is b_i, then log_(b_i)(b_i)=1 and the conversion formulae simplify. - The commonest bases
for logarithms are:
- The common logarithm
with base
10 and
function
symbol log(x).
This is the logarithms
usually used for
logarithmic plots
with
one power of
10 being called a
dex.
- The natural logarithm
with base
exponential e = 2.7182818 ...
and usually function
symbol ln(x).
The natural logarithm
is the logarithm
that appears naturally
in physics
formulae.
By "naturally" we mean it gives the simplest or most natural form.
Note:
Y_natural = [log_e(10)/log_e(e)]*y_common = [log_10(10)/log_(10)(e)]*y_common Y_natural = (2.302585 ...)*y_common = (1/0.43429448 ...)*y_common .

Note ln(x) can be pronounced "lawn x", "el-en x", or "natural log x" (see 2.5.1 The Natural Logarithm). Yours truly says "lawn x". - The logarithm
with base
2 with
function
symbol log_(2)(x).
This logarithm has some
special uses.
- The astronomical magnitude
with generic symbol M
(though other symbols are used as needed)
is a logarithm of
brightness (e.g.,
luminosity, or
related quantities) with funny
base and a wrong-way direction.
Say f is brightness, then the
astronomical magnitude
satisfies
f = f_0*100**(-M/5) log(f) = -(M/5)*log(100) + log(f_0) = -2.5*M + log(f_0) M = -0.4*log(f/f_0) ,

where 10 logarithms are used, f_0 is some reference brightness, and log(100) = 2*log(10) = 2. Thus an increase/decrease in astronomical magnitude by 5 gives a decrease/increase in f by a factor of 100. The base (which is almost never explicilty used isb = 100**(1/5) = 10**(2/5) = 10**(0.4) = 2.511886431509580111 ... ≅ 2.512 .

The fact that the astronomical magnitude is a wrong-way logarithm function with a funny base is very irritating. However, the astronomical magnitude system originated in classical antiquity and was regularized in the 19th century. It should have been discarded in the 19th century. But astronomy is the oldest empirical science, and so we are stuck with astronomical magnitude system which is used for reporting luminosities---and yes, brighter stars have lower astronomical magnitudes.

- The common logarithm
with base
10 and
function
symbol log(x).
This is the logarithms
usually used for
logarithmic plots
with
one power of
10 being called a
dex.