What is the domain of $y=\log _5 x$?
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 (2)
The domain of a logarithmic function lo g b x is all positive real numbers.
The function is y = lo g 5 x .
Therefore, the domain of y = lo g 5 x is 0"> x > 0 .
The domain is all real numbers greater than 0, which can be written as 0}"> x > 0 .
Explanation
Understanding the Problem The problem asks us to find the domain of the function y = lo g 5 x . The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Logarithmic Functions Logarithmic functions are only defined for positive arguments. This means that the argument of the logarithm, which is x in this case, must be greater than 0.
Determining the Domain Therefore, the domain of y = lo g 5 x is all real numbers greater than 0.
Final Answer The domain of the function y = lo g 5 x is 0"> x > 0 .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of solutions (pH). Understanding the domain of a logarithmic function is crucial in these contexts because it ensures that the input values are meaningful and the results are valid. For example, you can't have a negative earthquake intensity or a negative sound loudness, so the input to the logarithmic scale must be positive.
The domain of the function y = lo g 5 x consists of all real numbers greater than 0, which is represented mathematically as 0"> x > 0 . Therefore, the correct answer is option B. all real numbers greater than 0.
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