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Mirrors > Home > ILE Home > Th. List > df-logb | GIF version |
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 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.) |
Ref | Expression |
---|---|
df-logb | ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clogb 13199 | . 2 class logb | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 7709 | . . . 4 class ℂ | |
5 | cc0 7711 | . . . . 5 class 0 | |
6 | c1 7712 | . . . . 5 class 1 | |
7 | 5, 6 | cpr 3557 | . . . 4 class {0, 1} |
8 | 4, 7 | cdif 3095 | . . 3 class (ℂ ∖ {0, 1}) |
9 | 5 | csn 3556 | . . . 4 class {0} |
10 | 4, 9 | cdif 3095 | . . 3 class (ℂ ∖ {0}) |
11 | 3 | cv 1331 | . . . . 5 class 𝑦 |
12 | clog 13116 | . . . . 5 class log | |
13 | 11, 12 | cfv 5163 | . . . 4 class (log‘𝑦) |
14 | 2 | cv 1331 | . . . . 5 class 𝑥 |
15 | 14, 12 | cfv 5163 | . . . 4 class (log‘𝑥) |
16 | cdiv 8524 | . . . 4 class / | |
17 | 13, 15, 16 | co 5814 | . . 3 class ((log‘𝑦) / (log‘𝑥)) |
18 | 2, 3, 8, 10, 17 | cmpo 5816 | . 2 class (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) |
19 | 1, 18 | wceq 1332 | 1 wff logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) |
Colors of variables: wff set class |
This definition is referenced by: rplogbval 13201 |
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