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Mirrors > Home > MPE Home > Th. List > logbchbase | Structured version Visualization version GIF version |
Description: Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.) |
Ref | Expression |
---|---|
logbchbase | ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐴 logb 𝑋) = ((𝐵 logb 𝑋) / (𝐵 logb 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4726 | . . . . 5 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0)) | |
2 | logcl 25765 | . . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (log‘𝑋) ∈ ℂ) | |
3 | 1, 2 | sylbi 216 | . . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ) |
4 | 3 | 3ad2ant3 1135 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ) |
5 | logcl 25765 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | |
6 | 5 | 3adant3 1132 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ∈ ℂ) |
7 | logccne0 25775 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ≠ 0) | |
8 | 6, 7 | jca 513 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → ((log‘𝐴) ∈ ℂ ∧ (log‘𝐴) ≠ 0)) |
9 | 8 | 3ad2ant1 1133 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((log‘𝐴) ∈ ℂ ∧ (log‘𝐴) ≠ 0)) |
10 | logcl 25765 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (log‘𝐵) ∈ ℂ) | |
11 | 10 | 3adant3 1132 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (log‘𝐵) ∈ ℂ) |
12 | logccne0 25775 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0) | |
13 | 11, 12 | jca 513 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0)) |
14 | 13 | 3ad2ant2 1134 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0)) |
15 | divcan7 11726 | . . 3 ⊢ (((log‘𝑋) ∈ ℂ ∧ ((log‘𝐴) ∈ ℂ ∧ (log‘𝐴) ≠ 0) ∧ ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0)) → (((log‘𝑋) / (log‘𝐵)) / ((log‘𝐴) / (log‘𝐵))) = ((log‘𝑋) / (log‘𝐴))) | |
16 | 4, 9, 14, 15 | syl3anc 1371 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (((log‘𝑋) / (log‘𝐵)) / ((log‘𝐴) / (log‘𝐵))) = ((log‘𝑋) / (log‘𝐴))) |
17 | eldifpr 4597 | . . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) | |
18 | logbval 25957 | . . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) | |
19 | 17, 18 | sylanbr 583 | . . . 4 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) |
20 | 19 | 3adant1 1130 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵))) |
21 | 17 | biimpri 227 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1})) |
22 | eldifsn 4726 | . . . . . . 7 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | |
23 | 22 | biimpri 227 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ (ℂ ∖ {0})) |
24 | 23 | 3adant3 1132 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → 𝐴 ∈ (ℂ ∖ {0})) |
25 | logbval 25957 | . . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝐴) = ((log‘𝐴) / (log‘𝐵))) | |
26 | 21, 24, 25 | syl2anr 598 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) → (𝐵 logb 𝐴) = ((log‘𝐴) / (log‘𝐵))) |
27 | 26 | 3adant3 1132 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝐴) = ((log‘𝐴) / (log‘𝐵))) |
28 | 20, 27 | oveq12d 7321 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((𝐵 logb 𝑋) / (𝐵 logb 𝐴)) = (((log‘𝑋) / (log‘𝐵)) / ((log‘𝐴) / (log‘𝐵)))) |
29 | eldifpr 4597 | . . . 4 ⊢ (𝐴 ∈ (ℂ ∖ {0, 1}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1)) | |
30 | logbval 25957 | . . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐴 logb 𝑋) = ((log‘𝑋) / (log‘𝐴))) | |
31 | 29, 30 | sylanbr 583 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐴 logb 𝑋) = ((log‘𝑋) / (log‘𝐴))) |
32 | 31 | 3adant2 1131 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐴 logb 𝑋) = ((log‘𝑋) / (log‘𝐴))) |
33 | 16, 28, 32 | 3eqtr4rd 2787 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐴 logb 𝑋) = ((𝐵 logb 𝑋) / (𝐵 logb 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 ∖ cdif 3889 {csn 4565 {cpr 4567 ‘cfv 6454 (class class class)co 7303 ℂcc 10911 0cc0 10913 1c1 10914 / cdiv 11674 logclog 25751 logb clogb 25955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-inf2 9439 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 ax-addf 10992 ax-mulf 10993 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-of 7561 df-om 7741 df-1st 7859 df-2nd 7860 df-supp 8005 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-er 8525 df-map 8644 df-pm 8645 df-ixp 8713 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-fsupp 9169 df-fi 9210 df-sup 9241 df-inf 9242 df-oi 9309 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-7 12083 df-8 12084 df-9 12085 df-n0 12276 df-z 12362 df-dec 12480 df-uz 12625 df-q 12731 df-rp 12773 df-xneg 12890 df-xadd 12891 df-xmul 12892 df-ioo 13125 df-ioc 13126 df-ico 13127 df-icc 13128 df-fz 13282 df-fzo 13425 df-fl 13554 df-mod 13632 df-seq 13764 df-exp 13825 df-fac 14030 df-bc 14059 df-hash 14087 df-shft 14819 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 df-limsup 15221 df-clim 15238 df-rlim 15239 df-sum 15439 df-ef 15818 df-sin 15820 df-cos 15821 df-pi 15823 df-struct 16889 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-mulr 17017 df-starv 17018 df-sca 17019 df-vsca 17020 df-ip 17021 df-tset 17022 df-ple 17023 df-ds 17025 df-unif 17026 df-hom 17027 df-cco 17028 df-rest 17174 df-topn 17175 df-0g 17193 df-gsum 17194 df-topgen 17195 df-pt 17196 df-prds 17199 df-xrs 17254 df-qtop 17259 df-imas 17260 df-xps 17262 df-mre 17336 df-mrc 17337 df-acs 17339 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-submnd 18472 df-mulg 18742 df-cntz 18964 df-cmn 19429 df-psmet 20630 df-xmet 20631 df-met 20632 df-bl 20633 df-mopn 20634 df-fbas 20635 df-fg 20636 df-cnfld 20639 df-top 22084 df-topon 22101 df-topsp 22123 df-bases 22137 df-cld 22211 df-ntr 22212 df-cls 22213 df-nei 22290 df-lp 22328 df-perf 22329 df-cn 22419 df-cnp 22420 df-haus 22507 df-tx 22754 df-hmeo 22947 df-fil 23038 df-fm 23130 df-flim 23131 df-flf 23132 df-xms 23514 df-ms 23515 df-tms 23516 df-cncf 24082 df-limc 25071 df-dv 25072 df-log 25753 df-logb 25956 |
This theorem is referenced by: (None) |
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