4.4 The Natural Logarithm Function

1. 4.4 The Natural Logarithm Function

2. If we examine the reflection of points the line y = x, we will observe the following correspondence

3. Let us now consider the graph of y = ex.

4. If we observe the reflections of the points of this exponential function, we see the graph to the right

5. For each positive x, there is exactly one value of y such that (x,y) is on the new (reflected) graph. We call this value of y the natural logarithm of x and denote it as ln x. So, the reflection of the graph of y = ex through the line y = x is the graph of the natural logarithm function y = ln x.

6. Some properties of the natural logarithm are: • The point (1,0) is on the graph of y = ln x [because (0,1) is on the graph of y = ex]. So,ln 1 = 0. • ln x is defined only for positive values of x. • ln x is negative for x between 0 and 1. • ln x is positive for x greater than 1. • ln x is an increasing function and concave down.

7. We note that the point (a,b) is on the graph of ln x if and only if (b, a) is on the graph of ex. A typical point on the graph of ln x has the form (a, ln a), a > 0. So, for any positive value of a, the point (ln a, a) is on the graph of ex. This tells us that or more generally,

8. For each positive number x, ln x is the exponent to which we must raise e in order to get x. Now consider that if b is any number, then eb is positive so ln(eb) makes sense. Since (b, eb) is on the graph of ex, (eb,b) must be on the graph of ln x. This tells us that ln(eb) = b. In general,

9. We conclude by recalling that functions of the form bx can be written in terms of the function ex when b is a positive constant. We can now explicitly state that and so