Sequences of digits in decimal representation of powers of two?

Question

Mathematics

Actually, the question is: for any sequence of decimal digits, is it possible to specify such a power of two, in the decimal expansion of which it will occur? And one more thing: how exactly can such an exponent be calculated? I am also interested in all sorts of materials related to the issue.
01234567 found in... 2 ⁹⁴⁸⁵=185998256324337487767985099955302490665374438073335840272084270397484327790005864934072354482021720911607439665630887454808841855527882667163771398720576027402669580169039741850993940761617266622811864429881040671809439489622393109013165551200850833380933507325558571505564624577391929733959618246144977241913309659022912797548399276211570534729020491071351191290889636126433307695189713099137483570601661708273284507717023825388442891120735386162916399105393565702147204919204927037121724215287196788904608056905947536130449597358949449616051831907986767175594619583766525463617329127144503518763659942277269038905459230927989661987285329625187574824769288742763270245087547284539662518984216055365623621696956211603615545283516573789156694657671008997040572989022389295699721799750778506603701184419481263673489664211033837715782585254973569027982341874579420953999266744249299673279075763479994927498175002546738750955162901979077216015752598528464831465168854992680865829997423875909270885910965729572350830237045369449776277174455720770568210691006297077476042483939326526851979307090750800061956550603556501270224750819094942213629118099909177325337246786615531361534246491968360262811812215298071568437104822874348012370836228121589822780044987730811293226136550271694963934186396759877143359774015238471586730990536556332744717405897925895675069765677185890136024054302280347515023301882013138735177325337246786615531361534246491968360262811812215298071568437104822874348012370836228121589822780044987730811293226136550271694963934186396759877143359774015238471586730990536556332744717405897925895675069765677185890136024054302280347515023301882013138735177325337246786615531361534246491968360262811812215298071568437104822874348012370836228121589822780044987730811293226136550271694963934186396759877143359774015238471586730990536556332744717405897925895675069765677185890136024054302280347515023301882013138735012345674536443843500724073312200844070436544768321565730644790934123595660003672194718494935796078574313569457541209323393484644272286905883957862224606143980132378110678144627933700780795461464687395574955151452186658576874935551479315961297064331122140793650748872970125010994615841776971001836551719980645754836193958872457983938052160551666214065684852303128150087444204890759057273173850029313963600115272088506980655473444588000112953467252441483580477594551140995244830508723363302819687941270117760525826220041826619135638584326558176247505992357026743425616103447388318167511850110735269149887857412828805900122675014371794107967305858958751351419788686382412392518412100273742588751822095524644476110440217857585827426547924865922486611141726436838559414195948110593139374785498409121941376751557771017832079619769999600051786051653222771712610512883889823479670319462550051224474732291051859034097656799845251470736812586543469328564796307157930906773579236658091879601251498511749997235491825330900010269048713324079960509461271746921684816569470723524805341479469183318587968969248084022164275378401952871363267680996550172515684287413252616824865750445523554101027217128922653049387983014376322390036509760328751564360406787681572404381406365944853711323115460993534346396975045219006137336014139706260295262175593003673203123366570882309801070554543869201683365878375131751294372042353132928077347566058488076052266521358672510301765632

Answer 1

Hint:
n = log _a a ⁿ
a = b ^{log _b a}
Next, use row operations.
Start the proof for short sequences. Then, using deduction, find patterns in the number of possible degrees, etc.

Answer 2

Let's narrow down the problem: we will look for a power of two that starts with the desired sequence of digits (if the sequence starts from zero, we add one to the beginning). Let's denote it as d. Then we need to find x and y such that

d < 2 ^x /10 ^y < d + 1.

Let a = log ₂ 10. Then

d < 2x ^{- ay} < d + 1.

Logarithm to base 2:

log ₂ d < x - ay < log ₂ (d + 1)

It remains to choose x and y. Since a is irrational, it is obvious that we can find them.