What is the inverse of f(x)=5^4^x

Answer 1

y = 5^4^x

is ambiguous

it could mean (5^4)^x, if so then

y = 625^x, then switch x and y and solve for the new y

x = 625^y

log(625)x = y. exactly

y = logx to the base 625

y = lnx/ln625 = about .155334lnx= about .16lnx

or it could mean y=(5)^(4^x)

switch x and y

x = 5^(4^y) solve for the new y

logx = (4^y)log5

4^y = logx/log5

take logs again

ylog4 = log(logx/log5)

divide by log4

y = log(logx/log5)/log4 = f^-1(x) = inverse of f(x)=(5)^(4^x)

see video of Frank T., the other response to this problem. he assumes y=(5)^(4^x) as the problem

while most people will interpret the problem like Frank T did, if you're using calculators they can go either way depending on how they were programmed to deal with exponents of exponents

Answer 2

See the video.