Define L : (0, ) R by a) Prove that L is differentiable and strictly increasing on (0, ), with L'(x) = 1/x and

Define L : (0, ˆž) †’ R by

a) Prove that L is differentiable and strictly increasing on (0, ˆž), with L'(x) = 1/x and L(l) = 0.
b) Prove that L(x) †’ ˆž as x †’ ˆž and L(x) †’ -ˆž as x †’ 0+. (You may wish to prove

for all n ˆˆ N.)
c) Using the fact that {xq)' = qxq-1 for x > 0 and q ˆˆ Q (see Exercise 4.2.7), prove that L(xq) = qL(x) for all q ˆˆ Q and x > 0.
d) Prove that L(xy) = L(x) + L(y) for all x, y, ˆˆ (0, ˆž).
e) Suppose that e = limn†’ˆž + (1 + 1/n)n exists. (It does-see Example 4.22.) Use l'Hopital's Rule to show that Lie) = 1. [L(JC) is the natural logarithm function log x.]

L(x) = dt (2" ) = 2k-1 t !