Graph Of Y = Log_(1/2) X Explained

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In the realm of mathematics, understanding logarithmic functions is crucial for navigating various mathematical concepts and real-world applications. One such function that often presents a challenge is the logarithmic function with a fractional base. This article delves into the intricacies of the graph y = log_(1/2) x, providing a comprehensive exploration to aid in its understanding. We will dissect the function, explore its properties, and ultimately determine which of the given options accurately represents the same graph. Our journey will involve unraveling the relationship between logarithmic and exponential forms, and visualizing how the fractional base impacts the graph's behavior.

Understanding the Core of Logarithmic Functions

To effectively decipher the graph of y = log_(1/2) x, it's imperative to first establish a strong understanding of the fundamental principles governing logarithmic functions. In its essence, a logarithm answers the question: "To what power must we raise the base to obtain a specific number?" This contrasts with exponential functions, which ask: "What is the result of raising the base to a given power?"

The logarithmic function y = log_b x is defined as the inverse of the exponential function x = b^y, where 'b' represents the base, and 'x' and 'y' are variables. The base 'b' plays a pivotal role in determining the function's behavior, and it must adhere to specific constraints. The base must be a positive number, not equal to 1. This constraint is essential for maintaining the function's unique properties and preventing ambiguities in the result.

When b > 1, the logarithmic function y = log_b x is an increasing function. This means that as 'x' increases, 'y' also increases. Conversely, when 0 < b < 1, the logarithmic function becomes a decreasing function, where 'y' decreases as 'x' increases. The function y = log_(1/2) x falls under this latter category, as the base (1/2) is a fraction between 0 and 1. This is a crucial observation, as it immediately tells us the function will exhibit a downward trend as we move from left to right on the graph.

Another crucial aspect of logarithmic functions is their domain and range. The domain of a logarithmic function y = log_b x is the set of all positive real numbers, which means 'x' must always be greater than 0. This limitation arises because we cannot raise a positive base to any power and obtain a non-positive number. The range, on the other hand, encompasses all real numbers, implying that 'y' can take any value, positive, negative, or zero. Understanding these domain and range restrictions is essential for accurately interpreting and sketching the graph of the function.

Deconstructing y = log_(1/2) x: A Step-by-Step Analysis

Now, let's focus our attention specifically on the function at hand: y = log_(1/2) x. This logarithmic function has a base of 1/2, which, as we discussed earlier, means it will exhibit a decreasing behavior. To further dissect this function, we can leverage the fundamental relationship between logarithmic and exponential forms. The equation y = log_(1/2) x can be rewritten in its equivalent exponential form as x = (1/2)^y. This transformation provides a different perspective on the function and allows us to gain deeper insights into its graphical representation.

The exponential form x = (1/2)^y reveals the inverse relationship more explicitly. It shows how the value of 'x' changes as we vary 'y'. When 'y' is a large positive number, (1/2)^y becomes a very small positive number, approaching zero. As 'y' decreases towards zero, (1/2)^y approaches 1. When 'y' becomes negative, (1/2)^y becomes increasingly large. This is because (1/2)^-y is equivalent to 2^y, which grows exponentially as 'y' becomes more negative.

To visualize the graph, we can consider a few key points. When y = 0, x = (1/2)^0 = 1. This gives us the point (1, 0), which is the x-intercept of the graph. When y = 1, x = (1/2)^1 = 1/2. This gives us the point (1/2, 1). When y = -1, x = (1/2)^-1 = 2. This gives us the point (2, -1). By plotting these points and considering the decreasing nature of the function, we can begin to sketch the graph. It will start high on the left, approaching the y-axis as 'x' approaches 0, and then gradually decrease as 'x' increases, crossing the x-axis at (1, 0) and continuing downwards. The graph will never actually touch the y-axis, as the function is undefined for x = 0.

Evaluating the Options: Which Graph Matches y = log_(1/2) x?

Now that we have a solid understanding of the graph of y = log_(1/2) x, let's evaluate the given options and determine which one represents the same relationship.

A. x = 2^y

This equation represents an exponential function with a base of 2. As 'y' increases, 'x' also increases. This behavior is the inverse of what we expect from the graph of y = log_(1/2) x. Therefore, this option is incorrect.

B. x = -(2^y)

This equation also represents an exponential function, but with a negative sign in front. As 'y' increases, 2^y increases, but due to the negative sign, 'x' becomes increasingly negative. This graph would lie entirely in the left half of the coordinate plane, which is not consistent with the domain of logarithmic functions (x > 0). Therefore, this option is incorrect.

C. x = 2^(-y)

This equation can be rewritten as x = (2-1)y or x = (1/2)^y. This is precisely the exponential form equivalent to y = log_(1/2) x. Therefore, this option correctly represents the same graph.

Conclusion: The Equivalence of Logarithmic and Exponential Forms

In conclusion, the graph of y = log_(1/2) x is the same as the graph of x = 2^(-y). This equivalence highlights the crucial relationship between logarithmic and exponential functions. By understanding how to convert between these forms, we can gain a deeper understanding of the behavior and properties of these functions. The ability to manipulate and interpret logarithmic and exponential expressions is essential for success in various mathematical and scientific fields. This detailed exploration of y = log_(1/2) x serves as a valuable example of how to approach and analyze logarithmic functions, paving the way for a stronger grasp of more complex mathematical concepts.

By meticulously examining the function, its properties, and its relationship to the exponential form, we have successfully identified the correct representation. This exercise underscores the importance of a thorough understanding of mathematical principles and the power of applying those principles to solve problems. The ability to analyze and interpret logarithmic functions is a valuable skill, and this article provides a solid foundation for further exploration in the world of mathematics.