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What is the difference among the following:

$\log^²n , \log n^², \log\log n, (\log n)^²$
log n^2  can also be written as 2 log n

By default base of $\log = 10$. i.e.,

$\log 100 = 2 \implies {10}^2 = 100$

For base $e$ we use $\ln$ and for base $2$ we use $\lg.$

Now, coming to question, lets take $n=1000.$

1. $\log^2 n = \log n \times \log n = 3 \times 3 = 9.$
2. $\log n^2 = \log 1000000 = 6.$
3. $\log \log n = \log 3 = 0.4771.$
4. $(\log n)^2 = \log^2 n = 9.$

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1. log²n =(log n)2    : Here square is on logn not on n.

2. logn² : Here square is on n.So first calculate nthan take the log.

So   log²n =(log n) logn2

Consider n=100

Assuming base to be 10

log²n = (log n)2 = (log 100)2 = (2)2  =4

logn² = log1002 = log10000 = 4

loglogn =loglog100 =log(log100)=log2

(log n)2 = (log 100) = (2) =4

To understand the relation between log & exponent,consider

ax=y

Now,taking log on both sides,

xloga=logy

i.e. x=logay

So, all the above relations in the query can be obtained by replacing values for x, y & a.