PreCalculus

13.3. Logarithmic functions

The logarithm function for a positive real base b and a positive real number x is written as:

f(x) = log_b x

Note that we cannot take the logarithm of zero or a negative number.

The logarithm is defined as the inverse of raising the base b to the power x. That is,

y = log_b x

This is the inverse of:

x = b^y

Natural logarithm

The natural logarithm is the logarithm to the natural exponential base e; an irrational number (see above). By convention, the following notation is used for the natural logarithm function:

f(x) = ln x

Common logarithm

Similarly, the common logarithm is defined as logarithm base 10. By convention, it is written without indicating the base as:

f(x) = log x

Base 2 logarithm

A third convention, not as frequently encountered as the preceding, is logarithm to the base 2 which is used in computer science. It is written as:

f(x) = lg x

The above conventions are not always followed. Advanced mathematics texts frequently use the notation log x to mean the natural logarithm and write common logarithms as log₁₀ x.

To illustrate the effect of the selection of the base b, here is a table of values for the common, natural, and base 2 logarithms:

x         log x      ln x       lg x
--------------------------------------
0.125    -0.9031    -2.0794    -3.0000
0.250    -0.6021    -1.3863    -2.0000
0.500    -0.3010    -0.6931    -1.0000
1.000     0.0000     0.0000     0.0000
2.000     0.3010     0.6931     1.0000
3.000     0.4771     1.0986     1.5850
4.000     0.6021     1.3863     2.0000
5.000     0.6990     1.6094     2.3219
6.000     0.7782     1.7918     2.5850
7.000     0.8451     1.9459     2.8074
8.000     0.9031     2.0794     3.0000

Properties of logarithm functions

For the logarithm function: f(x) = log_b x

The domain of f(x) is: (0, +∞); its range is: (-∞, +∞)

The function is continuous, increasing, and is one to one.

A function must have one value of y for each value of x. A one to one function has one value of x for each value of y. This condition must be met for a function to have an inverse.

The graph of f(x) is concave downward.

0 = log_b 1

1 = log_b b

If log_b x = log_b y then x = y

The graph y = log_b (1/x) is symmetric to the graph of y = log_b x about the x-axis.

For example:

 x        ln x          ln (1/x)
--------------------------------
 0.1     -2.302585      2.302585
 0.2     -1.609437      1.609438
 0.5     -0.693147      0.693147
 1.0      0.000000      0.000000
 2.0      0.693147     -0.693147
 5.0      1.609438     -1.609437
10.0      2.302585     -2.302585