ln 20*log 5 * log e*log 2/log 20/log 25*log 100/log 16?

After simplifying all the multiplications and divisions, you get:

Numerator:  ln(20) * log(5) * log(e) * log(2) * log(100)

Denominator:  log(20) * log(25) * log (16)

The change of base formula says that ln(20) = log(20) / log(e)

         so log(e) / log(20)  =  1 / ln(20)

         so ln(20) * [log(e) / log(20)]  becomes ln(20) * [1 / ln(20) ]  =  1

Now, the numerator that we have left is:   log(5) *  log(2) * log(100)

the denominator is:                                 log(25) * log(16)

Now, log(100)  =  2   and    log(25)  =  log(5^2)  =  2log(5)    and    log(16)  =  log(2^4)  =  4log(2)

Making these substitutions, we have:

Numerator:  log(5) * log(2) * 2

Denominator:  2 * log(5) * 4 * log(2)

Cancelling, we end with:  1/4

ln 20*log 5 * log e*log 2/log 20/log 25*log 100/log 16 =  5/4