For a logarithmic spiral given parametrically as
(1)
 
(2)

evolute is given by
(3)
 
(4)

As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,
In some cases, the evolute is identical to the original, as can be demonstrated by making the substitution to the new variable
(5)

Then the above equations become
(6)
 
(7)
 
(8)
 
(9)

which are equivalent to the form of the original equation if
(10)

(11)

(12)

where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.
1  0.2744106319...  
2  0.1642700512...  
3  0.1218322508...  
4  0.0984064967...  
5  0.0832810611...  
6  0.0725974881...  
7  0.0645958183...  
8  0.0583494073...  
9  0.0533203211...  
10  0.0491732529... 