Devin's Reflection

Definition of the Natural Logarithmic Function

-the natural logarithmic function is defined by: ln x = f1^x 1/t dt, x > 0.

the domain of the natural logarithmic function is the set of all positive real numbers.

Theorem 5.1 Properties of the Natural Logarithmic Function

The natural logarithmic function has the following properties.

1. The domain is (0, oo) and the range is (-oo,oo).

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

3. The graph is concave downward.

Domain of f(x) = ln x^2

First derivative f’ = ln 2x

Second derivative f’’ = ln 2

Using the definition of the natural logarithmic function, you can prove several important properties involving operations with natural logarithms. If you are already familiar with logarithms, you will recognize that these properties are characteristic of all logarithms.

Theorem 5.2 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