4.1: Logs and Derivatives

Definition of the Natural Logarithm

Recall that

$$\int x^n\, dx = \dfrac{x^{n+1}}{n+1} +C. \nonumber$$

What is

$$\int x^{-1} dx \text{?} \nonumber$$

Definition

For $x > 0$ we define

$$\int_1^x \dfrac{1}{t} dt = \ln x. \nonumber$$

$$\dfrac{d}{dx} \ln x = \dfrac{1}{x}. \nonumber$$

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)

$$\dfrac{d}{dx} \ln x^n = \dfrac{d}{dx} \int_1^{x^n} \dfrac{1}{t} dt \nonumber$$

$$= \dfrac{1}{x^n} \left( n\,x^{n-1}\right) = n \left(\dfrac{1}{x} \right) \nonumber$$

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

So that

$$\ln x^n \nonumber$$

and

$$n \ln x \nonumber$$

have the same derivative. Hence

$$\ln x^n = n \ln x + C. \nonumber$$

Plugging in $x = 1$ we have that $C = 0$.

Definition of $e$

Let $e$ be such that

$$\ln e = 1 \nonumber$$

ie.

$$\int_1^e \dfrac{1}{t} dt = 1. \nonumber$$

Example 1

Find the derivative of

$$\ln (x^2 + 1). \nonumber$$

Solution

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

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

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.