LOGARITHMIC FUNCTIONS
Section: Calculus
 
1. EXPLANATION. COMMON AND NATURAL LOGARITHMS

Logarithmic functions are those functions in the form f(x) = loga(x) where "a" is constant (a number) and is called the base of the logarithm.

The most commonly used logarithmic functions in mathematics are "natural (or Naperian) logarithms", which are usually written ln (x), (Common logarithms to base 10 are usually written log(x)).

The window below illustrates both of these logarithmic functions explained above.
"Note that in the windows natural logarithms are referred to as log(x) and common logarithms  log10(x)"
1.- See if you remember the following definitions and if not write them down in your exercise book.

You will probably already know that the definition of the logarithm of (b) to the base (a) is: loga(b)=n when an=b.

We can therefore define the logarithm of a number "b" to a certain base "a" as the exponent to which we must raise the base a to get the number b. 

This is the relationship between exponential and logarithmic functions. Later on we will see what their graphs look like.

We also know that the base (a) of a logarithm must be a positive number (just like the base number of an exponential function) and cannot be 1 as in general log1(b) does not exist, as if b is not 1,1n cannot be b.

We also know that most logarithms are to base 10 (common logarithms) and base "e" = 2.718 281.. (natural or Naperian logarithms).
2. LOGARITHMS TO ANY BASE

The following window illustrates the logarithmic function in any base y = loga(x)
1.- Look at how the values of "y" alter when the x values are changed (Change them in the corresponding box at the bottom of the window).

Simply reduce the scale with the red "zoom" button to see more values on the screen. Although these values do not appear on the graph they are indicated in the top left-hand part of the screen.

2.- Now do the same for the values of "a". What do you notice?

3.- Look in particular at the logarithms of 2, 4, 8, 16 etc to base 2. Or look at the logarithms of 10, 100, 1,000 etc to base 10 (common logarithms).
4.- In particular, note that the function does not exist when a is negative or when a = 1, as mentioned above.

5.- Look carefully at the difference between the functions when a>1 or when a<1. Certain changes occur like they did with exponential functions! 

6.- Work out the value of the unknown letter in the following examples. Use the rules of logarithms and the window to help you.

a) log3 9 = k; .. b) log1/2 8 = k; .

c) loga 625 = 4; d) loga 1 = 0 ; 

e) log10 x = 4 ; f) log3 x = 2

We can use these observations to deduce some initial conclusions about logarithmic functions.

-The following must be true for the function to exist and to be able to draw its graph:  a > 0 and a # 1.
3. LOGARITHMIC FUNCTIONS AND EXPONENTIAL FUNCTIONS (THEIR INVERSE)

The following window shows the logarithmic function to the base "a" and the exponential function to the same base.

1.- Note the values of both functions when the value of x is altered.

2.- For example, for base a = 2 and x = 2, the exponential function is 4 and the logarithm is 1, whilst for example, when x = -1 and a = 2, the exponential function is 0.5 and the logarithm does not exist.

Exponential and logarithmic functions are defined as the inverse of each other. However, you may not be familiar with the concept of the inverse function. We can see that their graphs are symmetrical about the straight line y = x, as illustrated in this window.
4. GENERAL PROPERTIES

From now take the following to be true: a > 0 y que a # 1 . This window illustrates the properties.

1.- Note that the function only exists for values of x which are greater than 0, as opposed to the exponential function which exists for any value of x. (Use the basic rules of logarithms to see that the logarithm of a negative number or of 0 does not exist).

Therefore:

The DOMAIN of a logarithmic function is R+ or the interval (0, infinity)

2.- Use the basic rules of logarithms to prove numerically that log0(a), log2(-3), log1/2(-4) and in general loga(b), do not exist when b is negative. Look carefully at the graphs in the windows.
3.- Note that all functions always go through the fixed point (1,0) (make the value of x = 1 in the window to see the value of y). Therefore the graph always:

CUTS THE X-AXIS at the point (1,0).

4.- Note that when a>1 the curve gets closer and closer to the Y-axis as it moves downwards, without actually cutting it, and the same thing happens when a<1 as it moves upwards ("ALWAYS TOWARDS THE RIGHT"). Therefore:

THE Y-AXIS IS A VERTICAL ASYMPTOTE
       
           
  Leoncio Santos Cuervo
 
Spanish Ministry of Education. Year 2001
 
 

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