Definition df-logb 13656

Description: Define the log_b operator. This is the logarithm generalized to an arbitrary base. It can be used as ( B log_b X ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition will only be useful where x is a positive real apart from one and where y is a positive real, so the choice of ( CC \ { 0 , 1 } ) and ( CC \ { 0 } ) is somewhat arbitrary (we adopt the definition used in set.mm). (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

Step	Hyp	Ref	Expression
1		clogb 13655	. 2 class logb
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cc 7772	. . . 4 class CC
5		cc0 7774	. . . . 5 class 0
6		c1 7775	. . . . 5 class 1
7	5, 6	cpr 3584	. . . 4 class { 0 , 1 }
8	4, 7	cdif 3118	. . 3 class ( CC \ { 0 , 1 } )
9	5	csn 3583	. . . 4 class { 0 }
10	4, 9	cdif 3118	. . 3 class ( CC \ { 0 } )
11	3	cv 1347	. . . . 5 class y
12		clog 13571	. . . . 5 class log
13	11, 12	cfv 5198	. . . 4 class ( log ` y )
14	2	cv 1347	. . . . 5 class x
15	14, 12	cfv 5198	. . . 4 class ( log ` x )
16		cdiv 8589	. . . 4 class /
17	13, 15, 16	co 5853	. . 3 class ( ( log ` y ) / ( log ` x ) )
18	2, 3, 8, 10, 17	cmpo 5855	. 2 class ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
19	1, 18	wceq 1348	1 wff logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )

This definition is referenced by:  rplogbval  13657