Chapter 6 Elementary Functions

Section 6.4 Exponential and Logarithmic Functions

6.4.4 Logarithmic Function

In Section 6.4.3 we studied the natural exponential function

g:  { (0;) xg(x)=ex

In particular, we pointed out a very important property of the natural exponential function: it is strictly increasing. If the graph of this function is reflected about the angle bisector between the first and the third quadrant (see Chapter 9), one obtains the graph of the natural logarithmic function that has the symbol ln:
Info 6.4.7
The function defined by the equation eln(x) =x

ln:  { (0;) xln(x)

is called the natural logarithmic function.

Here, the equation shall be read in such a way that ln(x)=a is just the value a with ea =x. The construction mentioned above is shown in the figure below.

The following properties of the natural logarithmic function can be seen from the graph:
  • The function ln is strictly increasing.
  • Approaching zero from the right on the x-axis, ln(x) takes ever larger negative values: We note that the graph of ln gets arbitrarily close to the negative vertical axis ( y-axis).
  • At the point x=1 the natural logarithmic function takes the value 0, i.e. ln(1)=0.

As well as the natural logarithmic function there are other logarithmic functions, which each correspond to a certain exponent.
Info 6.4.8
If b>0 is an arbitrary exponent, then the function defined by the equation b logb (x) =x (read as: " logb (x)=a is the exponent a with ba =x")

logb :  { (0;) x logb (x)

is called the general logarithmic function with base b.

Normally, the logarithmic function cannot be calculated directly. Since it is defined as the inverse function of the exponential function, one generally tries to rewrite its input as a power and reads off the exponent.
Example 6.4.9
Typical calculations for the natural logarithmic function are

ln(e5 )  =  5,ln(e)  =  ln(e 1 2 )  =   1 2

and for the general logarithmic function:

log5 (25)  =   log5 ( 52 )  =  2, log3 (81)  =   log3 ( 34 )  =  4.

Here, the base of the logarithmic function has to be observed, for example, we have

log2 (64)  =   log2 ( 26 )  =  6,   but    log4 (64)  =   log4 ( 43 )  =  3.

Exercise 6.4.10
Calculate the values of the following logarithmic functions:
  1. ln(e3) =
  2. log2 (256) =
  3. log9 (3) =

In mathematics and in the sciences the following logarithmic functions are frequently used and thus have their own dedicated names:
  • Logarithmic function to the base 10: denoted by log10 (x)= lg (x) or sometimes only by log(x). This logarithmic function is associated to the powers of ten and is used, for example, in chemistry for the calculation of the pH level.
  • Logarithmic function to the base 2: denoted by log2 (x)= ld (x). This logarithmic function is relevant in computer science.
  • Logarithmic function to the base e: denoted by loge (x)=ln(e). The natural logarithmic function is mostly inadequate for practical calculations (unless the expression is a power of e). It is called natural since the exponential function with base e, from a mathematical point of view, is simpler than the general exponential function (e.g. because ex is its own derivative, but bx for be is not).

There are several calculation rules for the logarithmic function, which will be explained in the next section.