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Definition df-logb 23406
Description: Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as  (logb `  <. B ,  X >. ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". You could also use  ( Blogb X ), which looks like a less-common notation that some use where the base is a preceding superscript. Note: This definition doesn't prevent bases of 1 or 0; proofs may need to forbid them. (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 23405 . 2  class logb
2 vx . . 3  set  x
3 vy . . 3  set  y
4 cc 8751 . . . 4  class  CC
5 cc0 8753 . . . . 5  class  0
6 c1 8754 . . . . 5  class  1
75, 6cpr 3654 . . . 4  class  { 0 ,  1 }
84, 7cdif 3162 . . 3  class  ( CC 
\  { 0 ,  1 } )
95csn 3653 . . . 4  class  { 0 }
104, 9cdif 3162 . . 3  class  ( CC 
\  { 0 } )
113cv 1631 . . . . 5  class  y
12 clog 19928 . . . . 5  class  log
1311, 12cfv 5271 . . . 4  class  ( log `  y )
142cv 1631 . . . . 5  class  x
1514, 12cfv 5271 . . . 4  class  ( log `  x )
16 cdiv 9439 . . . 4  class  /
1713, 15, 16co 5874 . . 3  class  ( ( log `  y )  /  ( log `  x
) )
182, 3, 8, 10, 17cmpt2 5876 . 2  class  ( x  e.  ( CC  \  { 0 ,  1 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( ( log `  y
)  /  ( log `  x ) ) )
191, 18wceq 1632 1  wff logb  =  ( x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
Colors of variables: wff set class
This definition is referenced by:  logbval  23407
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