The Natural Logarithmic Function Differentiation

Definition of the Natural Logarithmic Function The natural logarithmic function is defined by The domain of the natural logarithmic function is the set of all positive real numbers

Properties of the Natural Logarithmic Function The domain is (0, ∞) and the range is (- ∞, ∞). The function is continuous, increasing, and one-to-one. The graph is concave downward.

Graph of a the Natural Logarithmic Function

Logarithmic Properties If a and b are positive numbers and n is rational, then the following properties are true. 1. ln (1) = 0 2. ln(ab) = ln a + ln b 3. ln(a n ) = n ln a 4. ln (a/b) = ln a – ln b

Properties of Logarithms Use the properties of logarithms to approximate ln 0.25 given that ln 2 ≈ and ln 3 ≈ (b) ln 24 (c) ln 1/72

Expanding Logarithmic Expressions Use the properties of logarithms to expand the logarithmic expression

Logarithms as a Single Quantity Write the expression as a logarithm of a single quantity (a) 3 ln x + 2 ln y – 4 ln z (b) 2 ln 3 - ½ln (x 2 + 1) (c) ½[ln (x 2 + 1) – ln (x + 1) – ln (x – 1)]

The Number e The base of the natural logarithmic function is e e ≈

Definition of e The letter e denotes the positive real number such that

Evaluating Natural Logarithmic Expressions ln2 ln 32 ln 0.1

Derivative of the Natural Logarithmic Function In other words, the derivative of the function over the function.

Differentiation of Logarithmic Functions Find the derivative of the function (a) h(x) = ln (2x 2 + 1) (b) f(x) = x ln x

Differentiation of Logarithmic Functions

Logarithmic Properties as Aids to Differentiation

More Examples P. 322 problems 60 On-line Examples

Logarithmic Differentiation

P. 322 problems 87 – 92 On-line Examples

Finding the Equation of the Tangent Line Find an equation of the tangent line to the graph of f at the indicated point

Locating Relative Extrema Locate any relative extrema and inflection points for the graph of Y = x – ln x Y = lnx/x