1. ## Logarithm integral question....

Define the natural logarithm function in terms of the area under the curve y=1/x.
I need some help getting this one started. I also need to explain how to use this definition to find the derivative of y= ln x, and explain how to use the derivative formula from above to prove that ln(a) + ln(b)= ln(ab)... I know it's a lot, but I could really use a push in the right direction. Thank you for the push...

2. Originally Posted by bemidjibasser
Define the natural logarithm function in terms of the area under the curve y=1/x.
I need some help getting this one started. I also need to explain how to use this definition to find the derivative of y= ln x, and explain how to use the derivative formula from above to prove that ln(a) + ln(b)= ln(ab)... I know it's a lot, but I could really use a push in the right direction. Thank you for the push...
The area under a curve f(x) is defined as the integral f(x) dx

If $\displaystyle f(x) = \frac{1}{x}$

Then the area under the curve is:

$\displaystyle \int \frac{1}{x} dx$

$\displaystyle = ln |x|$

The antiderivative of $\displaystyle \frac{1}{x}$ is $\displaystyle ln|x|$