DITF - Natural Logarithmic Function and Differentiation Lesson

Natural Logarithmic Function and Differentiation

Natural Logarithmic Function and its Graph

graph of logarithm The natural logarithm of a positive number x, written as ln x, is the value of an integral. The natural logarithmic function f(x) = ln x, with e as the base of the logarithm, is defined by equation image indicator $LaTeX: \ln x=\int_1^x\frac{1}{t}dt$ $\ln x=\int_1^x\frac{1}{t}dt$ . The domain of the natural logarithmic function is the set of all positive real numbers. Geometrically, if x > 1, then ln x is the area under the continuous curve y=1/t from t = 1 to t = x. This graph reinforces the idea of the natural logarithmic function as the antiderivative given by the differential equation equation image indicator $LaTeX: \frac{dy}{dx}=\frac{1}{x}$ $\frac{dy}{dx}=\frac{1}{x}$ .

Properties of Natural Logarithmic Function

By the defining properties of the natural logarithmic function, equation image indicator $LaTeX: y=\ln x\Leftrightarrow e^y=x$ $y=\ln x\Leftrightarrow e^y=x$ . In addition, $LaTeX: \ln e^x=x,\:x\epsilon\mathbb{R}$ $\ln e^x=x,\:x\epsilon\mathbb{R}$ and $LaTeX: e^{\ln x}=x,\:x>0$ $e^{\ln x}=x,\:x>0$ . Note that when x = 1, ln e = 1. The natural logarithmic function has the same properties as general logarithmic functions text annotation indicator .

1. The domain is equation image indicator $LaTeX: \left(0,\infty\right)$ $\left(0,\infty\right)$ and the range is $LaTeX: \left(-\infty,\infty\right)$ $\left(-\infty,\infty\right)$ .

2. The function is continuous, increasing, and one-to-one.

3. The graph is concave downward and passes through the point (1, 0).

Other properties that are algebraic in nature are characteristic of all logarithms. These properties of logarithms are sometimes referred to as laws of logarithms.

Logarithmic Properties Valid for any positive numbers x and y and any exponent n
Product Rule	$LaTeX: \ln xy=\ln x+\ln y$ $\ln xy=\ln x+\ln y$
Quotient Rule	$LaTeX: \ln\frac{x}{y}=\ln x-\ln y$ $\ln\frac{x}{y}=\ln x-\ln y$
Reciprocal Rule	$LaTeX: \ln\frac{1}{x}=-\ln x$ $\ln\frac{1}{x}=-\ln x$
Power Rule	$LaTeX: \ln x^n=n\:\ln x$ $\ln x^n=n\:\ln x$

View the presentation on properties of logarithms and their application to simplifying logarithmic expressions.

Derivative of the Natural Logarithmic Function

Recall that y = ln x written in logarithmic notation is equivalent to e^y = x written in exponential notation. The identity e^{ln x} = x is a useful result of this relationship. Differentiating the exponential form implicitly with respect to x is the basis for deriving a formula for the derivative of the natural logarithmic function.

$LaTeX: e^y=x\\ e^y\frac{dy}{dx}=1\\ \frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}$ $e^y=x\\ e^y\frac{dy}{dx}=1\\ \frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}$

View the presentation below on using the limit definition of a derivative to find the derivative of the natural logarithmic function.

If u is a differentiable function of x where x > 0, then applying the Chain Rule produces the general rule for differentiating y=ln u.

equation image indicator $LaTeX: \frac{dy}{dy}=\frac{dy}{du}\cdot\frac{du}{dx}\\ \frac{d}{dx}\ln u=\frac{d}{du}\ln u\cdot\frac{du}{dx}=\frac{1}{x}\cdot\frac{du}{dx}\\ \frac{d}{du}\ln u=\frac{1}{u}\cdot\frac{du}{dx}\left(u>0\right)$ $\frac{dy}{dy}=\frac{dy}{du}\cdot\frac{du}{dx}\\ \frac{d}{dx}\ln u=\frac{d}{du}\ln u\cdot\frac{du}{dx}=\frac{1}{x}\cdot\frac{du}{dx}\\ \frac{d}{du}\ln u=\frac{1}{u}\cdot\frac{du}{dx}\left(u>0\right)$

Consider an interesting phenomenon related to the natural logarithmic function. The graph of y = f(ax) is a horizontal stretch (0 < a < 1) or shrink (a > 1) of the graph of y = f(x) by a factor of 1/a. The graph of f(x) + C is a vertical shift of the graph of y = f(x). By applying the product property of logarithms, ln ax = ln a + ln x. Since ln a is a constant, ln ax = ln x + C, which means the graph of y = ln ax is a vertical shift of the graph of y = ln x. This suggests that if the shapes of the two graphs are identical, their derivatives will be the same.

If u = ax, then

equation image indicator $LaTeX: \frac{d}{du}\ln u=\frac{1}{u}\cdot\frac{du}{dx}\left(u>0\right)\\ \frac{d}{du}\ln ax=\frac{1}{ax}\cdot\frac{d}{dx}\left(ax\right)=\frac{1}{ax}\cdot a\\ \frac{d}{du}\ln ax=\frac{1}{x}=\frac{d}{du}\ln x$ $\frac{d}{du}\ln u=\frac{1}{u}\cdot\frac{du}{dx}\left(u>0\right)\\ \frac{d}{du}\ln ax=\frac{1}{ax}\cdot\frac{d}{dx}\left(ax\right)=\frac{1}{ax}\cdot a\\ \frac{d}{du}\ln ax=\frac{1}{x}=\frac{d}{du}\ln x$

Logarithmic Differentiation

Finding derivatives of complicated functions involving products, quotients, or powers can be simplified by taking logarithms. The process of differentiating after taking logarithms of both sides of an identity is called logarithmic differentiation. The procedure for logarithmic differentiation is as follows:

1. Take logarithms of both sides of an equation y = f(x).

2. Use properties of logarithms to simplify.

3. Differentiate implicitly with respect to x.

4. Solve the resulting equation for y'.

The presentation below illustrates the use of logarithmic differentiation.

The next presentation is an extension of the differentiation of y = x^x.

Natural Logarithmic Function and Differentiation Practice

Natural Logarithmic Function and Differentiation: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the natural logarithmic function and differentiation.

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