Identifying the Logarithmic Function- Which of the Following is the Correct Choice-

Which of the following is a logarithmic function? This question often comes up in mathematics, particularly when discussing exponential and logarithmic functions. Understanding the characteristics of a logarithmic function is crucial for solving various mathematical problems and comprehending the relationships between different types of functions. In this article, we will explore the features of logarithmic functions and provide examples to help you identify them among other functions.

A logarithmic function is a mathematical function that is the inverse of an exponential function. It has the general form f(x) = log_b(x), where ‘b’ is the base of the logarithm and ‘x’ is the argument. The logarithmic function is used to find the exponent to which the base must be raised to obtain a given number. For instance, if we have the equation log_2(8) = 3, it means that 2 raised to the power of 3 equals 8.

One of the key features of a logarithmic function is its domain and range. The domain of a logarithmic function is all positive real numbers because the logarithm of a negative number or zero is undefined. The range of a logarithmic function depends on the base. If the base is greater than 1, the range is all real numbers. If the base is between 0 and 1, the range is negative infinity to positive infinity.

Another characteristic of a logarithmic function is its increasing or decreasing behavior. When the base is greater than 1, the logarithmic function is increasing, meaning that as the argument increases, the function’s value also increases. Conversely, when the base is between 0 and 1, the logarithmic function is decreasing, meaning that as the argument increases, the function’s value decreases.

To determine whether a given function is logarithmic, we can follow these steps:

1. Check if the function has the form f(x) = log_b(x), where ‘b’ is the base and ‘x’ is the argument.
2. Ensure that the base is greater than 0 and not equal to 1.
3. Verify that the domain of the function consists of all positive real numbers.

Let’s consider a few examples to illustrate the identification of logarithmic functions:

Example 1: f(x) = log_3(x)
This function is logarithmic because it has the form f(x) = log_b(x), with a base greater than 0 and not equal to 1, and its domain is all positive real numbers.

Example 2: f(x) = log_2(x^2)
This function is not logarithmic because it does not have the form f(x) = log_b(x). However, if we rewrite it as f(x) = 2 log_2(x), it becomes a logarithmic function with a base of 2 and a domain of all positive real numbers.

Example 3: f(x) = log_0.5(x)
This function is logarithmic because it has the form f(x) = log_b(x), with a base between 0 and 1, and its domain is all positive real numbers.

In conclusion, identifying a logarithmic function involves recognizing its form, base, and domain. By understanding the characteristics of logarithmic functions, you can solve various mathematical problems and appreciate the beauty of these functions in the world of mathematics.