What is the range of [tex]$y=\log _8 x$[/tex]?
A. all real numbers less than 0
B. all real numbers greater than 0
C. all real numbers not equal to 0
D. all real numbers
Answer (2)
The range of a function is the set of all possible output values.
The function y = lo g 8 x is the inverse of the exponential function x = 8 y .
The exponential function 8 y is always positive for any real number y .
Therefore, the range of y = lo g 8 x is all real numbers. all real numbers
Explanation
Understanding the Problem We are asked to find the range of the function y = lo g 8 x . The range of a function is the set of all possible output values (y-values) that the function can produce.
Relating Logarithmic and Exponential Functions Recall that the logarithmic function y = lo g b x is the inverse of the exponential function x = b y , where b is the base of the logarithm. In our case, the base is 8, so we have y = lo g 8 x , which is equivalent to x = 8 y .
Considering the Domain of the Logarithm We need to determine what values y can take. Since x must be greater than 0 for the logarithm to be defined, we have 0"> x > 0 . Thus, we need to find the values of y for which 0"> 8 y > 0 .
Analyzing the Exponential Function The exponential function 8 y is always positive for any real number y . As y approaches − ∞ , 8 y approaches 0, but never actually reaches 0. As y approaches ∞ , 8 y also approaches ∞ . Therefore, 8 y can take any positive value.
Determining the Range Since 8 y can take any positive value, x can be any positive number. This means that y can be any real number. To see this, consider any real number y . We can always find a positive value of x such that x = 8 y . Therefore, the range of y = lo g 8 x is all real numbers.
Final Answer The range of the function y = lo g 8 x is all real numbers.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding the range of a logarithmic function helps us interpret the possible values in these applications. For example, if we are using a logarithmic scale to measure sound intensity, knowing the range tells us the possible values of sound intensity that can be represented on the scale. Logarithmic scales are also used to represent data that spans a wide range of values, making it easier to visualize and analyze the data.
The range of the function y = lo g 8 x is all real numbers because for any real number y , there exists a corresponding positive value of x . Thus, the correct choice is D. all real numbers.
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