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.
For y = lo g 5 x , the argument x must be greater than 0.
Therefore, the domain is 0"> x > 0 .
The domain of y = lo g 5 x 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.
Logarithm Definition The logarithm function, denoted as lo g b x , is only defined for positive values of x . In other words, the argument of the logarithm must be greater than zero. The base b must be a positive number not equal to 1. In our case, the base is 5, which satisfies this condition.
Determining the Domain Therefore, for the function y = lo g 5 x to be defined, we must have 0"> x > 0 . This means that x can be any real number 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 tells us the valid range of input values for the model. For example, when modeling population growth, the input (time) must be a non-negative number, as negative time does not make sense in this context. Similarly, the argument of the logarithm in the Richter scale must be positive, as negative intensity is not physically meaningful.
The domain of the function y = lo g 5 x consists of all real numbers greater than 0. This is because logarithmic functions are defined only for positive arguments. Thus, the correct answer is option B.
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