Calculus question, PLEASE HELP WITH THIS, due tonight!!
1 answer:
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Note that 
, 
, 
, and so on. The function 
is continuous and increasing for all 
, so when 
, we have 
; when 
, 
; and so on.
This means
which means we can capture the number of digits of
with the function 
.
So the problem is the same as finding positive integer solutions to
We know that
has one digit, so clearly this must be a solution. We need to show that this is the only solution.
Recall that![\dfrac{\mathrm d}{\mathrm dx}[\log_{10}x+1]=\dfrac1{\ln10\,x}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B%5Clog_%7B10%7Dx%2B1%5D%3D%5Cdfrac1%7B%5Cln10%5C%2Cx%7D)
, while ![\dfrac{\mathrm d}{\mathrm dx}[x]=1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5D%3D1)
. This means increases at a much slower rate than 
as 
. We know the two functions intersect when 
. Therefore it's clear that 
for all 
.
Now, it's always the case that
, so we're essentially done:
which means there are no other solutions than.
This means
which means we can capture the number of digits of
So the problem is the same as finding positive integer solutions to
We know that
Recall that
Now, it's always the case that
which means there are no other solutions than