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Mathematics LibreTexts

4.1: Logs and Derivatives

Definition of the Natural Logarithm

Recall that

\[ \int x^n\, dx = \dfrac{x^{n+1}}{n+1} +C.\]

What is

\[ \int x^{-1} dx \text{?}\]

Definition

For \(x > 0\) we define

\[ \int_1^x \dfrac{1}{t} dt = \ln x.\]

Note

The Second Fundamental Theorem of Calculus tells us that

\[ \dfrac{d}{dx} \ln x = \dfrac{1}{x}.\]

Properties of \(\ln x\):

  1. \(\ln 1 = 0\)
  2. \(\ln (ab) = \ln a + \ln b\)
  3. \(\ln(a^n) = n \ln a\)
  4. \(\ln \left( \dfrac{a}{b} \right) = \ln a - \ln b\)

Proof of (3)  

\[\dfrac{d}{dx} \ln x^n = \dfrac{d}{dx} \int_1^{x^n} \dfrac{1}{t} dt\]

\[ = \dfrac{1}{x^n} \left( n\,x^{n-1}\right) = n \left(\dfrac{1}{x} \right)\]

\[ = n \dfrac{d}{dx} \int _1^x \dfrac{1}{t} dt = n\ln x.\]

So that 

\[ \ln x^n \]

and

\[ n \ln x \]

have the same derivative. Hence

\[ \ln x^n = n \ln x + C.\]

Plugging in \(x = 1\) we have that \(C = 0\).

Definition of \(e\)

Let \(e\) be such that

\[ \ln e = 1 \]

ie. 

\[ \int_1^e \dfrac{1}{t} dt = 1.\]

Example 1

Find the derivative of

\[ \ln (x^2  + 1).\]

Solution

We use the chain rule with \( y = \ln u\) so \(u = x^2 + 1 \),

\[ y'  =  (2x)(1/u)  =  \dfrac{2x}{x^2+1}.\]

Exercise

Find the derivatives of the following functions:

  • \(\ln (\ln x)\)

  • \( \dfrac{\ln x}{x} \)

  • \((\ln x)^2 \)

  • \(\ln (\sec x)\)   

  • \(\ln (\csc x) \) 

Exercise

  1. Show that \( y= 3 \ln x - 4 \) is a solution of the differential equation \(xy'' + y' = 0\).
  2. Find the relative extrema of \( x \ln x\).

  3. Find the equation of the tangent line to \(y = 3x^2 - \ln x \) at \((1,3)\).

  4. Find \(\dfrac{dy}{dx}\) for \( \ln (xy) + 2x^2 = 30 \).

Contributors

  • Integrated by Justin Marshall.