Definition df-logb 25820

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 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 ∈ (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 25819	. 2 class logb
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 10800	. . . 4 class ℂ
5		cc0 10802	. . . . 5 class 0
6		c1 10803	. . . . 5 class 1
7	5, 6	cpr 4560	. . . 4 class {0, 1}
8	4, 7	cdif 3880	. . 3 class (ℂ ∖ {0, 1})
9	5	csn 4558	. . . 4 class {0}
10	4, 9	cdif 3880	. . 3 class (ℂ ∖ {0})
11	3	cv 1538	. . . . 5 class 𝑦
12		clog 25615	. . . . 5 class log
13	11, 12	cfv 6418	. . . 4 class (log‘𝑦)
14	2	cv 1538	. . . . 5 class 𝑥
15	14, 12	cfv 6418	. . . 4 class (log‘𝑥)
16		cdiv 11562	. . . 4 class /
17	13, 15, 16	co 7255	. . 3 class ((log‘𝑦) / (log‘𝑥))
18	2, 3, 8, 10, 17	cmpo 7257	. 2 class (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
19	1, 18	wceq 1539	1 wff logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))

This definition is referenced by:  logbval  25821  logbmpt  25843  logbfval  25845