Logarithmic Functions

Logarithmic Functions Definition

In mathematics, a logarithmic function is a function that is defined by an equation of the form

f(x) = log b (x)

In this equation, b is a real number (the base), and x is a positive real number (the argument). The function log b (x) is the inverse of the function bx. That is, if y = log b (x), then x = by.

Introduction to Common and Natural Logarithmic Function

There are two types of logarithmic functions: common logarithms and natural logarithms.

The common logarithm of a number is the logarithm to the base 10. The natural logarithm of a number is the logarithm to the base e.

The common logarithm of a number can be found using the following formula:

log 10 (x) = y

The natural logarithm of a number can be found using the following formula:

log e (x) = y

The following graph shows the common and natural logarithm functions:

Common Logarithmic Function

A logarithmic function is a function in which the output is based on the logarithm of the input. The most common type of logarithmic function is the natural logarithm, which is denoted by the symbol “ln.” The natural logarithm is used to measure the rate of change of a given quantity.

Natural Logarithmic Function

The natural logarithmic function is the inverse of the natural exponential function. It is written as:

,

where is the natural logarithm of .

The natural logarithmic function is used to find the power to which a number must be raised to get a given result. It is also used to find the inverse of a natural exponential function.

Properties of Logarithmic Functions

The domain of a logarithmic function is all real numbers.

The range of a logarithmic function is all real numbers.

A logarithmic function is increasing if the base is positive and decreasing if the base is negative.

A logarithmic function is inverse to an exponential function if the base is the same.

The Zero Exponent Rule

The zero exponent rule states that if a number is raised to the power of zero, the result is one.

For example,

5 × 5 × 5 × 5 × 5 = 5,000

5 raised to the power of zero is equal to 1.

Change of Base Rule

The change of base rule states that when a number is raised to a power, the power can be computed by multiplying the number by the base and then taking the root of the product.

For example, the cube root of 64 can be computed as follows:

64 × 3 = 192

192 ÷ 3 = 64

Examples of Logarithmic Functions

The natural logarithm function, denoted as ln(x), is a logarithmic function that is defined as the inverse of the exponential function, ex.

ln(x) = y

if and only if

y = ex

The following are some examples of logarithmic functions.

ln(x) = y

if and only if

y = 10x

ln(x) = y

if and only if

y = x/10

ln(x) = y

if and only if

y = 100x

How to Learn Logarithmic Functions?

There are a few different ways that you can learn logarithmic functions. One way is to use a calculator to help you understand the different properties of logarithmic functions. Another way is to use a table of logarithms to help you find the inverse of a logarithmic function. Finally, you can also use a graphing calculator to help you graph logarithmic functions.

Importance of Learning Logarithmic Functions

Logarithmic functions are important in many areas of mathematics, including calculus and differential equations. In addition, they are used in physics, engineering, and other sciences.