Determining the Range of a Logarithmic Function: A Comprehensive Guide

Understanding the range of a function is essential for comprehending its behavior and limitations. In this article, we delve into the process of determining the range of the function ( y log_5 (4x - x^2) ). This involves several steps, including analyzing the domain of the logarithmic function, solving a quadratic inequality, and finding critical points using the properties of quadratic functions.

Step-by-Step Guide

1. Domain of the Logarithmic Function

To find the range of ( y log_5 (4x - x^2) ), we first need to determine the domain of the expression inside the logarithm. This expression, ( 4x - x^2 ), must be positive because the logarithm of a non-positive number is undefined. We start by solving the inequality:

( 4x - x^2 > 0 )

We can rewrite this as:

( -x^2 4x > 0 )

This is a standard form of a quadratic equation ( ax^2 bx c > 0 ), where ( a -1 ), ( b 4 ), and ( c 0 ). To solve this, we find the roots of the corresponding quadratic equation:

( -x^2 4x 0 )

Using the quadratic formula, ( x frac{-b pm sqrt{b^2 - 4ac}}{2a} ), we get:

( x frac{-4 pm sqrt{4^2 - 4(-1)(0)}}{2(-1)} frac{-4 pm sqrt{16}}{-2} frac{-4 pm 4}{-2} )

Therefore, the roots are:

( x frac{0}{-2} 0 ) and

( x frac{-8}{-2} 4 )

Since the quadratic expression opens downwards (the coefficient of ( x^2 ) is negative), ( -x^2 4x ) is positive between the roots ( 0 ) and ( 4 ). Therefore, the domain of the function ( y log_5 (4x - x^2) ) is:

( 0

2. Finding the Maximum Value

To find the maximum value of ( 4x - x^2 ) within the interval ( [0, 4] ), we need to find the vertex of the quadratic function. The x-coordinate of the vertex for a quadratic function ( ax^2 bx c ) is given by:

( x -frac{b}{2a} )

Substituting the coefficients ( a -1 ) and ( b 4 ), we get:

( x -frac{4}{2(-1)} 2 )

Thus, the function reaches its maximum value at ( x 2 ). We find this value by substituting ( x 2 ) into the expression ( 4x - x^2 ):

( 4(2) - (2)^2 8 - 4 4 )

Therefore, the maximum value of ( 4x - x^2 ) within the interval ( [0, 4] ) is 4.

3. Logarithmic Transformation

Now, we need to determine the range of ( y log_5 (4x - x^2) ). Since the function ( 4x - x^2 ) has a maximum value of 4 and its minimum value within the domain is 0 (approaching 0 as ( x ) approaches the boundaries 0 and 4), we can write:

( 0

The logarithm function is increasing, so:

( log_5 (0^ ) -infty ) and

( log_5 (4) approx log_5 (5^{0.861}) 0.861 )

Thus, the range of ( y log_5 (4x - x^2) ) is:

( y in (-infty, log_5 (4)] ) or

( y in (-infty, 0.861] )

In conclusion, the range of the function ( y log_5 (4x - x^2) ) is:

( (-infty, 0.861] )

3. Key Takeaways

The domain of a logarithmic function must be positive.

The range of a logarithmic function based on a quadratic expression can be found by identifying the maximum and minimum values of the quadratic function.

The logarithmic function is increasing, so it transforms the maximum and minimum values of the quadratic function into specific range endpoints.