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Mathematics LibreTexts

4.1: Logs and Derivatives

  • Page ID
    530
  • [ "article:topic", "exponent", "natural logarithm", "authorname:green" ]

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    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 \( \ln (xy) + 2x^2 = 30 \).

    Larry Green (Lake Tahoe Community College)

    • Integrated by Justin Marshall.