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| Description: Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.) | 
| Ref | Expression | 
|---|---|
| logbval | ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵)) | |
| 2 | 1 | oveq2d 7448 | . 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵))) | 
| 3 | fveq2 6905 | . . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋)) | |
| 4 | 3 | oveq1d 7447 | . 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵))) | 
| 5 | df-logb 26809 | . 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥))) | |
| 6 | ovex 7465 | . 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V | |
| 7 | 2, 4, 5, 6 | ovmpo 7594 | 1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 {csn 4625 {cpr 4627 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 0cc0 11156 1c1 11157 / cdiv 11921 logclog 26597 logb clogb 26808 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-logb 26809 | 
| This theorem is referenced by: logbcl 26811 logbid1 26812 logb1 26813 elogb 26814 logbchbase 26815 relogbval 26816 relogbcl 26817 relogbreexp 26819 relogbmul 26821 nnlogbexp 26825 relogbcxp 26829 cxplogb 26830 logbgt0b 26837 dvrelog2b 42068 dvrelogpow2b 42070 rege1logbrege0 48484 logb2aval 49338 | 
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