What is the range of $y=\log _8 x$?
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 (1)
The problem asks for the range of the function y = lo g 8 x .
Recognize that y = lo g 8 x is equivalent to x = 8 y .
Since x must be positive, we analyze the possible values of y that make 8 y positive.
8 y is positive for all real numbers y , so the range of y = lo g 8 x is all real numbers.
The range of y = lo g 8 x is 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).
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 . In our case, the base b 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 positive for the logarithm to be defined (i.e., 0"> x > 0 ), we need to consider what values of y will make 8 y positive.
Analyzing the Exponential Function Since 8 is a positive number, 8 y will always be 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, x can take any positive value.
Determining the Range Since y can be any real number, the range of 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 loudness of sound (decibels), and modeling population growth or decay. Understanding the range of a logarithmic function helps in interpreting the possible values in these applications. For example, in seismology, the Richter scale uses logarithms to measure the magnitude of earthquakes. The range of the logarithmic function helps us understand that earthquake magnitudes can theoretically take any real value, although in practice, there are physical limits to how large or small an earthquake can be.
