Natural Logarithm - Properties

Properties

(see complex logarithm)

Proof
The statement is true for, and we now show that for all, which completes the proof by the fundamental theorem of calculus. Hence, we want to show that (Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity and subtracting we would obtain This statement is trivially true for since the left hand side is negative or zero. For it is still true since both factors on the left are less than 1 (recall that ). Thus this last statement is true and by repeating our steps in reverse order we find that for all . This completes the proof.

Proof

The statement is true for, and we now show that for all, which completes the proof by the fundamental theorem of calculus. Hence, we want to show that

(Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity and subtracting we would obtain

This statement is trivially true for since the left hand side is negative or zero. For it is still true since both factors on the left are less than 1 (recall that ). Thus this last statement is true and by repeating our steps in reverse order we find that for all . This completes the proof.

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