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Definition df-logb 23606
Description: Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as  ( B logb  X ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition is according to Wikipedia "Complex logarithm": https://en.wikipedia.org/wiki/Complex_logarithm#Logarithms_to_other_bases (10-Jun-2020). (Contributed by David A. Wheeler, 21-Jan-2017.)
Assertion
Ref Expression
df-logb  |- logb  =  (
x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-logb
StepHypRef Expression
1 clogb 23605 . 2  class logb
2 vx . . 3  setvar  x
3 vy . . 3  setvar  y
4 cc 9526 . . . 4  class  CC
5 cc0 9528 . . . . 5  class  0
6 c1 9529 . . . . 5  class  1
75, 6cpr 3995 . . . 4  class  { 0 ,  1 }
84, 7cdif 3430 . . 3  class  ( CC 
\  { 0 ,  1 } )
95csn 3993 . . . 4  class  { 0 }
104, 9cdif 3430 . . 3  class  ( CC 
\  { 0 } )
113cv 1436 . . . . 5  class  y
12 clog 23408 . . . . 5  class  log
1311, 12cfv 5592 . . . 4  class  ( log `  y )
142cv 1436 . . . . 5  class  x
1514, 12cfv 5592 . . . 4  class  ( log `  x )
16 cdiv 10258 . . . 4  class  /
1713, 15, 16co 6296 . . 3  class  ( ( log `  y )  /  ( log `  x
) )
182, 3, 8, 10, 17cmpt2 6298 . 2  class  ( x  e.  ( CC  \  { 0 ,  1 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( ( log `  y
)  /  ( log `  x ) ) )
191, 18wceq 1437 1  wff logb  =  (
x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  logbval  23607  logbmpt  23629  logbfval  23631
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