(this is most of it)Calculus
Lesson
26
Ln(x) and Differentiation
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First of all, recall everything that we know about logs:
(this is most of it)
Now, why is base e so important? Well, here is one definition of ln(x):
So, ln(x) is really the area under the curve f(t)=1/t between 1 and the t-value
you're looking at. Note that this makes sense according to what we know about
the logarithm!
Domain: (0, Infinity) Range: (-Infinity, Infinity)
Continuous
Concave Down
ln(1) = 0 (from FTC)
Now we also have geometric explanations of some of the other things we know:
ln(e)=1 means the area under the curve 1/t between t=1 and t=e is 1.
Domain must be only positive numbers since less than zero means we would
have tried to find an area through an infinite discontinuity.
Range goes negative since we are traveling in the negative direction from 1.
Also notice that from the 2nd FTC we can find: 
.
By the chain rule, 
Also, show that 
note
we don't have to worry about zero and
is referring to the
area under the curve of 
.
Examples: 
Logarithmic Differentiation:
then differentiate implicitly
On to Lesson 27 - Ln(x) and Integration