Define the natural logarithm of a number x in terms of an integral.Show that
ln(xy) = ln(x) + ln(y)
for all x,y >0
No it is not. How is defined? In fact, is defined to be the inverse function of but you never demonstrated it.
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Here is how to prove it. Let . Define a function 0,\infty)\mapsto \mathbb{R}" alt="f0,\infty)\mapsto \mathbb{R}" /> as . This function is differenciable and furthermore, . Thus, for some number . In particular, but . Thus, . Therefore, .