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 function is y = lo g 8 x , which is equivalent to x = 8 y .
The domain of the logarithm function is 0"> x > 0 .
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, so the answer is all real numbers .
Explanation
Understanding the Problem We are asked to find the range of the function y = lo g 8 x . This means we want to determine all possible values of y that the function can output.
Relating Logarithms and Exponentials Recall that the logarithm function is the inverse of the exponential function. In this case, y = lo g 8 x is equivalent to x = 8 y .
Considering the Domain Since x must be greater than 0 (the domain of the logarithm function is 0"> x > 0 ), we need to consider what values of y will make 8 y positive.
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 ∞ .
Determining the Range Therefore, y can be any real number. The range of y = lo g 8 x is all real numbers.
Examples
Logarithmic functions are used to model many real-world phenomena, such as the Richter scale for earthquake magnitudes, the pH scale for acidity, and the decibel scale for sound intensity. Understanding the range of a logarithmic function helps us interpret the possible values in these models. For example, the Richter scale uses a base-10 logarithm to quantify the magnitude of earthquakes. The scale is open-ended, meaning there is no theoretical upper limit to the magnitude of an earthquake, reflecting the fact that the range of the logarithmic function is all real numbers. Similarly, in finance, logarithmic scales are used to analyze investment growth, where the range helps in understanding potential returns.
The range of the function y = lo g 8 x is all real numbers because for every real number y , there exists a positive x = 8 y . There are no limitations on the values of y , confirming that the function can output any real number. Hence, the answer is all real numbers .
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