What is the domain of [tex]$y=\log _5 x$[/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 domain of a logarithmic function y = lo g b x is all positive real numbers.
For the function y = lo g 5 x , the argument x must be greater than 0.
Therefore, the domain is all real numbers greater than 0.
The domain of y = lo g 5 x is 0"> x > 0 , so the answer is all real numbers greater than 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.
Logarithm Definition Logarithmic functions are only defined for positive arguments. This means that the input to the logarithm must be greater than zero. In other words, for y = lo g 5 x to be defined, we must have 0"> x > 0 .
Determining the Domain Therefore, the domain of the function y = lo g 5 x is all real numbers greater than 0.
Final Answer The domain of y = lo g 5 x is all real numbers greater than 0.
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 domain of a logarithmic function is crucial in these applications because it ensures that the input values are valid and the results are meaningful. For example, you cannot have a negative earthquake intensity or a pH value that is not defined.
The domain of the function y = lo g 5 x is all real numbers greater than 0. Therefore, the correct choice is option B. all real numbers greater than 0.
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