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 (ℂ ∖ {0, 1}) and (ℂ ∖ {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 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {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 ℂ
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 (ℂ ∖ {0, 1})
9	5	csn 3583	. . . 4 class {0}
10	4, 9	cdif 3118	. . 3 class (ℂ ∖ {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 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
19	1, 18	wceq 1348	1 wff logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))

This definition is referenced by:  rplogbval  13657