Which statement is true?
A. [tex]$y=\log _{10} x$[/tex] is not a logarithmic function because the base is greater than 0.
B. [tex]$y=\log _{\sqrt{3}} x$[/tex] is not a logarithmic function because the base is a square root.
C. [tex]$y=\log _1 x$[/tex] is not a logarithmic function because the base is equal to 1.
D. [tex]$y=\log _{\frac{3}{4}} x$[/tex] is not a logarithmic function because the base is a fraction.
Answer (2)
Logarithmic function y = lo g b x requires the base b to be greater than 0 and not equal to 1.
Statement 1 is false because base 10 is valid.
Statement 2 is false because base 3 is valid.
Statement 3 is true because base 1 is invalid.
Statement 4 is false because base 4 3 is valid.
The true statement is: y = lo g 1 x is not a logarithmic function because the base is equal to 1 .
Explanation
Analyzing the Statements Let's analyze each statement to determine which one is true regarding logarithmic functions. A logarithmic function is defined as y = lo g b x , where b is the base, x is the argument, and b must be greater than 0 and not equal to 1 ( 0"> b > 0 and b = 1 ).
Evaluating Statement 1 Statement 1: y = lo g 10 x is not a logarithmic function because the base is greater than 0. The base is 10, which is greater than 0 and not equal to 1. Thus, y = lo g 10 x is a logarithmic function. The statement is false.
Evaluating Statement 2 Statement 2: y = lo g 3 x is not a logarithmic function because the base is a square root. The base is 3 , which is approximately 1.732. This is greater than 0 and not equal to 1. Thus, y = lo g 3 x is a logarithmic function. The statement is false.
Evaluating Statement 3 Statement 3: y = lo g 1 x is not a logarithmic function because the base is equal to 1. The base is 1, which violates the condition that the base cannot be equal to 1. Thus, y = lo g 1 x is not a logarithmic function. The statement is true.
Evaluating Statement 4 Statement 4: y = lo g 4 3 x is not a logarithmic function because the base is a fraction. The base is 4 3 , which is 0.75. This is greater than 0 and not equal to 1. Thus, y = lo g 4 3 x is a logarithmic function. The statement is false.
Conclusion Therefore, the true statement is: y = lo g 1 x is not a logarithmic function because the base is equal to 1.
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 in decibels, and modeling population growth. Understanding the constraints on the base of a logarithm is crucial for correctly applying these functions. For example, in chemistry, pH is defined using a logarithmic scale with base 10, where pH = -log10[H+], with [H+] being the concentration of hydrogen ions. If the base were allowed to be 1, the logarithmic scale would not be useful because log1(x) is undefined, making it impossible to define pH values accurately.
The true statement is C: y = lo g 1 x is not a logarithmic function because the base is equal to 1. The other statements are false as their bases either satisfy the logarithmic function conditions or are valid. Therefore, only Statement C is correct.
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