Natural Logarithm revisited

So recall we introduced the natural logarithm. Integrals interpret the natural logarithm as \int t^{-1}\,\mathrm{d}t, and we will explore that in this post.

1. The natural logarithm satisfies
(1)\displaystyle\frac{\mathrm{d}\ln(x)}{\mathrm{d}x}=\frac{1}{x}
By the fundamental theorem of calculus, we can consider a different definition
(2)\displaystyle\ln(x)=\int^{x}_{1}\frac{\mathrm{d}t}{t}
Observe that, by (1), the first derivative is positive for x\geq0. Also by differentiating it again, we get
(3)\displaystyle\frac{\mathrm{d}^{2}\ln(x)}{\mathrm{d}x^{2}}=\frac{-1}{x^{2}}
Thus \ln(x) is concave down but increasing. Thus its sketch should look like:

So What?!? Well, we have an effective method to calculate the natural logarithm of any number! As a matter of fact, we can show a doodle what the logarithm corresponds to:

The curve f(x)=1/x is drawn is blue, and the area between x=1 and x=4 is shaded in light blue. That light blue region’s area corresponds to the natural logarithm.

WARNING: The blue curve doodled is the derivative of the natural logarithm. This is a subtle point which some students miss, and believe the blue curve is the natural logarithm. This tempting mistake must be avoided!

2. What about \mathrm{e}? Recall that \mathrm{e} is the number satisfying
(4)\displaystyle\ln(\mathrm{e})=1
The integral form of this statement is
(4′)\displaystyle\int^{\mathrm{e}}_{1}\frac{\mathrm{d}t}{t}=1
We can doodle this situation as

We want the shaded green region to have an area equal to 1.

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About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Calculus, Definite Integral, Integral, Natural Logarithm. Bookmark the permalink.

2 Responses to Natural Logarithm revisited

  1. Pingback: [Index] Calculus of a Single Variable | My Math Blog

  2. Pingback: [Index] Calculus in a Single Variable | My Math Blog

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