Claim: $\log 5\notin\mathbb{Q}$: Proof: $10^{a/b}=5\implies 10^a=5^b$. But for positive integers $a,b$, we have $10^a$ is even and $5^b$ is odd. $\square$ Since $\log 5\notin\mathbb{Q}$, and the set $S=\{(\log 5^n)\mod 1|n\in\mathbb{N}\}=\{(n\log 5)\mod 1|n\in\mathbb{N}\}$ is dense in $[0,1]$. This means every interval $I\subset [0,1]$ contains the number $\log 5^n\mod 1$ for some $n\in\mathbb{N}$. Now just letting $I=[\log 2.022,\log 2.023)$, we know there is an $n$ that puts $\log 5^n\in I$. For that $n$ the number $5^n$ starts with $2022$. Fun fact: The smallest solution is $n=7470$ corresponding to $5^{7470}=$ \begin{align*} &2022704264342289870213869330044956034441093900279888007031211864755873\\& 7555471321534686436510361709547320785472844366909525091779956467181224\\& 1544044299737023133576894678666578815197533867472289760246931696430315\\& 3336414652252535225618539631579174758027974186174254354299592161786845\\& 6035674768009887448273345216187349870110471324110181245982841570102629\\& 5054927519858813312579006280736917749495082398535657220062397632041741\\& 6884981327382104468988297191326550590584376357346227373366297881635294\\& 6445688752965753462319549375821536341380298687426299663748912819512740\\& 7800342976399101012212961574985907636008536735195056506027767516356033\\& 7261653199354108992543891447669181552923935598512515377444949559701184\\& 1456538015325648350747058512123487233728914684391449600460568662580467\\& 3380083755886334936497416531220614452113411108265419834352120147723770\\& 5033027999457631090409318266547956067275258095607858687028657484290917\\& 8181822074098005825877467080107619138304703333428566142672274482233561\\& 3728044663919377254980680857324878041027494271940690746871996203478060\\& 7183928871931039285286128483374296775951938927773318658929135351075808\\& 5002749650361975407331329438434801947066625532739552971158597962607017\\& 3452967441362500878552864269310158518875120916047722741608064259282189\\& 6922394040449819996753881701391444678369575669655869153417237854488920\\& 5994858876702406586382164990927038055134055996642766473355600102117223\\& 4419705727047356799695446972003764078533886943755964452515625209165292\\& 6657953720927194613006200851899147011244649082574953400995911439553340\\& 8410639087475165653243924944537661843423637210916563889972058952701401\\& 9677362841894593725210164556725678852127582028723950526941052526613457\\& 8448510730117452179153843195536458222422562756872429333158921580439130\\& 2224869520264849020954874944071410668337275046876696878897994403960135\\& 0527151755029453084133032020735853443079254777611171255141064521005990\\& 0702495917709613727845733670947592572747531555487742513319277082456235\\& 3199912957646707586415283319862862149065034490119910759554626874327726\\& 7187608414455658294164256442547282973233179530701741091572725090327396\\& 5922243135328603049798014803212386268055423224528968743489861798374429\\& 4902438777746086655109707233384384849629515886289723136066164810144762\\& 0888284774501381026738280841292849594648260901575507763070683043304251\\& 3578514380816840625918599820685784248932080923868920712573657357058126\\& 0702827675287213331118631493756964869358428933127490589710125895923040\\& 3544495465882751253597893321419138947664746640360471689557018022913933\\& 0505959959720727713181069922389924066369368139436808913670732029185401\\& 3138075929370077044217970966606295462532347093192491792275362206951615\\& 7074561557985241982127777596930653470304741004834566044210109021439767\\& 7338044986721255294758480880841950669901621758350025237152908538549025\\& 7153816724765173756990034638435034700503726244517609197902541744846121\\& 1828869646859175880401149932609902508627112274612143482009071849898311\\& 4391428774307553897632802054149649352081410304000954855467966797652197\\& 5149641983742447257662083888827879620055748403962934055696467209532289\\& 6706839384228334724419551322391841418966297394504939324336423585184293\\& 4735358213820926954513265483889772809441342051129965245568169853074763\\& 7986814097350449613644346668100850987464879933688795081070082277353304\\& 4231813099832226068255015997580922493555421675851222311602383093702165\\& 5262122635506638120430794125644072128473012139142726498603524227533322\\& 6797572308713324896707852205246728455085550857175262638837472424051268\\& 2632005168936992339189186085657433438436783169100852875497837169989312\\& 6019126828798081891249523130412967486302775944167678351106208620390003\\& 3044366067747782085051252658970307861901743112083674828731126765838844\\& 0035996298000505837920416661811163660141958112796551209771320487580917\\& 3693967238588934177444137655060094303846434765387602720939652766453331\\& 0067120297963207070504518085197176388146964298773432343366541361392343\\& 2048122022576634996161407728562682276741638185717172891493216325166775\\& 3183561206178001158299723294734045431572215483372952941578514393920322\\& 5655570646750514791678025788834174992491045252404026074433011059867735\\& 5560921881168295355793529116827163349226122988106742399207343119654415\\& 1118978568100804616290412178960763136440398020559151834727935388928129\\& 1527652417248705141671129562041170293732655039356946263662874477112292\\& 5162692361326069933782785349388919197119305037334839270180698589104290\\& 6933572854754296666773397670052578385222979526501266275696430075903066\\& 8735470337511932393173577908923654945049131559843697711125425059046999\\& 2011493438959866043081881734708604802983034569393858787598448107116044\\& 6882782068608789977571476735336628631334766215010209418713359633237974\\& 8065378253857126539867803669959052360017375746586001409177601428867006\\& 9020837259044411375156785735556955608517017257136033140591179025239823\\& 2964637047956811571000635682334501217407005447706083861857596591327263\\& 6708728296986264273050499443575147376376611158324920035068131746886824\\& 4430119932243610631350440148679864672292859632769463925987037026993599\\& 1484889902275606120245277756049909569056153102121635282660635184904515\\& 7179990782295586038388697047837123564248583745000138866695782054240548\\& 321344395077403532923199236392974853515625 \end{align*}

OMGG, THANK YOU SO MUCH!!

Very nice...How to find the min value n=7470?

soryn wrote: Very nice...How to find the min value n=7470? I did it by running a small R-script using the GNU Multiple Precision library.

The number 2022 is (unsurprisingly) not special. We may prove that for any pair of positive integers $(k,N)$ such that $log(k)$ is irrational, there exist a natural number $n$ such that $k^n$ starts with the number $N$. Let $d$ be the number of digits of $N$. It`s not hard to prove that the number formed by the first $d$ digits of the number $k^n$ is given by the expresion $\lfloor k^n\cdot 10^{-\lfloor log(k^n)\rfloor +d} \rfloor=\lfloor 10^{\{ log(k)n \}+d} \rfloor$ (Recall $n=\lfloor n \rfloor +\{ n\}$) Note that the sequence $(\{ log(k)n \})_{n \geq 1}$ is dense in $(0,1)$, so $(10^{\{ log(k)n \}})_{n \geq 1}$ is dense in $(1,10)$. It follows that there exist a ntural number $n$ that satisfy the inequality $\frac{N}{10^{d-1}}<10^{\{ log(k)n \}}<\frac{N+1}{10^{d-1}}$. Such $n$ does the trick.

See the chapter 15 of Problems from the Book for harder applications of this idea