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.

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

  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) 

  log_b(x**p) = log_b([b**y]**p) = log_b(b**[py]) = py = p*log_b(x)  . 

  x = b_1**y_1  and  x = b_2**y_2  implying  b_1**y_1 = b_2**y_2  .  

  y_1*log_b(b_1) = y_2*log_b(b_2)  or  y_2 = [(log_b(b_1)/log_b(b_2)]*y_1  . 

1. 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.

2. 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".

3. The logarithm with base 2 with function symbol log_(2)(x). This logarithm has some special uses.

4. 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 is

     b = 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.