4.2
The Natural Logarithmic Functions
Definition 23 The natural logarithmic function is the function that asso-ciates with each positive real number the power to which must be raised to produce . This function is represented by
ln
So, if we write
= ln we mean that is such that
= From the definition, we have
ln = ⎧ ⎨ ⎩ 0 1; 0 1; = 0 = 1 ln 5 3.75 2.5 1.25 0 0 -5 -10 -15 -20 -25 -30 x y x y
Theorem 52 The natural logarithmic function has the following properties.
1. The domain is (0 ∞) and the range is (−∞ ∞) 2. The function is continuous, increasing, and one-to-one.
3. The graph is concave downward.
Proof. Use tha graph.
Theorem 53 If and are positive numbers and is rational, then the following properties are true.
1. ln (1) = 0; 2. ln () = ln + ln ; 3. ln¡¢= ln ; 4. ln¡¢ = ln − ln 5. ln = Example 100 a. ln√3 + 2 b. ln65 c. ln ( 2+3)2 3√2+1
Example 101 Solve equation 7 = +1 ( = ln 7
− 1) Example 102 Solve equation ln (2 − 3) = 5
Theorem 54 Suppose
ln () = +
where 6= 0 Then () is an exponential function that can be expressed in the form
() = where =