Lesson WHAT IS the logarithm

Logarithm

Definition of logarithm

The logarithm of a number x to a given base b is the power y to which the base must be raised to get the number x.
In mathematical symbols, .
An expression is read "the logarithm, base b, of x".

In expression the real number x is called an argument, while the real number b is called the base.
The logarithm is defined for all positive real numbers x. So, any positive real number x can be an argument.
The base of the logarithm can be any positive real number, except of 1.

Examples
1) . An argument of the logarithm is equal to '8' here, the base is equal to '2'. The value of is equal to 3 because .
2) . An argument of the logarithm is equal to '100' in this example, the base is equal to '10'. The value of is equal to 2 because .

It follows from the logarithm definition, that if , then .
Why? Simply, if , then the base value b raised to the degree y should be equal to x. This is exactly what expression states.
Actually, both these expressions, and , are equivalent.

The direct consequence of it is an equality , as well as an equality .

Examples for the last two formulas
3) ; 4) .

More examples of logarithms
5) . An argument of the logarithm is equal to 1/8 here, the base is equal to 2. The value of is equal to -3 because .
6) . An argument of the logarithm is equal to 0.01 in this example, the base is equal to 10. The value of is equal to -2 because .

You see that the logarithms themselves can be negative.


More examples of logarithms with the base value less than 1
7) . Here an argument of the logarithm is equal to 8, the base is equal to 1/2. The value of is equal to -3 because .
8) . In this example an argument of the logarithm is equal to 1/8, the base is equal to 1/2. The value of is equal to 3 because .

Logarithmic function

Let's consider the logarithmic function for the case when the base b of the logarithm is greater than 1, for example, for the base value b=2.
For values of x=1/4, 1/2, 1, 2, 4, 8 the corresponding values of logarithm are equal to -2, -1, 0, 1, 2, 3.
The plot of the logarithmic function is shown in Figure 1 below.
First, you see that for this case values of the logarithmic function are negative for values of x less than 1 and positive for values of x greater than 1.
You see also that the logarithmic function is monotonically increased when the argument x is increased.
This is the typical plot and the typical behavior of the logarithmic function for the case when the base is greater than 1.

Figure 1. Logarithmic function

Figure 2. Logarithmic function

Let's consider the logarithmic function for the case when the base b of the logarithm is less than 1, for example, for the base value b=1/2=0.5.
For values of x=1/4, 1/2, 1, 2, 4, 8 the corresponding values of logarithm are equal to 2, 1, 0, -1, -2, -3.
The plot of the logarithmic function is shown in Figure 2 above.
In contrast to the previous example, you see that for this case values of the logarithmic function are positive for values of x less than 1 and negative for values of x greater than 1.
You see also that the logarithmic function is monotonically decreased when the argument x is increased.
This is the typical plot and the typical behavior of the logarithmic function for the case when the base is less than 1.