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y=log(x+1) shaped curve where x=infinity -> y=1

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Which specifications does your function have?

Perhaps this will satisfy your needs.:

y=1 - 1/(x+1)

This function is 0 for x=0 and 1 for x=infinity

It is strictly monotonic increasing just as the logarithm.
If you need something with a log-Function you have to check the L'Hospital rule or sth.

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