solvarmedw

2022-09-01

Log of many Logs

How can I compute the values of $n$ for which the following expression exists?

$${\mathrm{log}}_{e}({\mathrm{log}}_{e}({\mathrm{log}}_{e}({\mathrm{log}}_{e}(\dots {\mathrm{log}}_{e}(n))))$$

It is for instance apparent that when $n=e$, the second application of ${\mathrm{log}}_{e}$ is undefined.

How can I compute the values of $n$ for which the following expression exists?

$${\mathrm{log}}_{e}({\mathrm{log}}_{e}({\mathrm{log}}_{e}({\mathrm{log}}_{e}(\dots {\mathrm{log}}_{e}(n))))$$

It is for instance apparent that when $n=e$, the second application of ${\mathrm{log}}_{e}$ is undefined.

Kaleb Harrell

Beginner2022-09-02Added 14 answers

As $\mathrm{ln}x$ is apriori defined only for $x>0$, you need

$$n>{e}^{{e}^{{e}^{{e}^{{e}^{0}}}}}$$

where the tower has as many $e$s as your expression has ${\mathrm{log}}_{e}$s

$$n>{e}^{{e}^{{e}^{{e}^{{e}^{0}}}}}$$

where the tower has as many $e$s as your expression has ${\mathrm{log}}_{e}$s

Gunsaz

Beginner2022-09-03Added 3 answers

You have the sequence

$$0,1,e,{e}^{e},{e}^{{e}^{e}},{e}^{{e}^{{e}^{e}}},\dots ,$$

for which for the $n$th member (starting with $0$), you can apply $\mathrm{log}$ only $n$ times before you end up with $\mathrm{log}0$. The sequence diverges, so there is no point at which you can take as many $\mathrm{log}$s as you like.

$$0,1,e,{e}^{e},{e}^{{e}^{e}},{e}^{{e}^{{e}^{e}}},\dots ,$$

for which for the $n$th member (starting with $0$), you can apply $\mathrm{log}$ only $n$ times before you end up with $\mathrm{log}0$. The sequence diverges, so there is no point at which you can take as many $\mathrm{log}$s as you like.

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

Which operation could we perform in order to find the number of milliseconds in a year??

$60\cdot 60\cdot 24\cdot 7\cdot 365$ $1000\cdot 60\cdot 60\cdot 24\cdot 365$ $24\cdot 60\cdot 100\cdot 7\cdot 52$ $1000\cdot 60\cdot 24\cdot 7\cdot 52?$ Tell about the meaning of Sxx and Sxy in simple linear regression,, especially the meaning of those formulas

Is the number 7356 divisible by 12? Also find the remainder.

A) No

B) 0

C) Yes

D) 6What is a positive integer?

Determine the value of k if the remainder is 3 given $({x}^{3}+k{x}^{2}+x+5)\xf7(x+2)$

Is $41$ a prime number?

What is the square root of $98$?

Is the sum of two prime numbers is always even?

149600000000 is equal to

A)$1.496\times {10}^{11}$

B)$1.496\times {10}^{10}$

C)$1.496\times {10}^{12}$

D)$1.496\times {10}^{8}$Find the value of$\mathrm{log}1$ to the base $3$ ?

What is the square root of 3 divided by 2 .

write $\sqrt[5]{{\left(7x\right)}^{4}}$ as an equivalent expression using a fractional exponent.

simplify $\sqrt{125n}$

What is the square root of $\frac{144}{169}$