You seem to be thinking of this problem as 1+(lne)^4, which would be 1+1^4=1+1=2, but really it means 1+ln(e^4)=1+4=5.

When you have the log (or natural log, in this case) of something to a power, let's say ln(X^Y), that's equal to Y*lnX. So for 1+ln(e^4), that equals 1+4*lne=1+4*1=5.

To look at it another way, lne=1, because the base (e) only has to be to the first power to equal e. For ln(e^2)=2 because the base (e) has to be squared to equal e^2. That's true for any number that e is raised to.

You can test what I've said above with log base 10 and various powers of ten if you want to see for yourself.

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