1848.sagews

Section 7.2*: The Natural Logarithmic Function

Recall:  The power rule for integration says:

$

\int x^n\ dx=\frac{x^{n+1}}{n+1}+C

$

as long as $n\neq -1$.

 

Definition:  The natural log function is the function defined by

$

\ln(x)=\int_1^x \frac{1}{t}\ dt

$

for $x>0$.

Recall that the graph of $y=\frac{1}{t}$ is as follows.

 

var('t')
plot(1/t,(t,.3,8))

Here is the picture of $y=\ln(x)$ for $x>1$.  The value of the area of the shaded region is equal to $\ln(7)$ in this case.

plot(1/t,(t,.3,8))+plot(1/t,(t,1,7),fill=true)

Here is the picture for $x<1$. 

plot(1/t,(t,.3,8))+plot(1/t,(t,1/2,1),fill=True)

Important:  In the case of $0<x<1$, the area under the curve is positive, but the value of, say $\ln(1/2)$ is negative.  Why?

 

$

\int_1^{1/2}1/t\ dt=-\int_{1/2}^11/t\ dt

$

What happens if $x=1$?

$

\ln(1)=\int_1^1 1/t\ dt=0.

$

 

The graph of $y=\ln(x)$ is as follows.  (Let's see if we can figure out what to type into Sage.)

plot(ln(x),(x,0,10))