To determine which statement among the given options is true, we need to understand the properties of logarithmic functions.
A logarithmic function [tex]\( y = \log_b{x} \)[/tex] is defined if and only if the base [tex]\( b \)[/tex] follows these conditions:
1. The base [tex]\( b \)[/tex] must be a positive real number.
2. The base [tex]\( b \)[/tex] cannot be equal to 1.
Let's evaluate each statement individually:
1. Statement 1: [tex]\( y = \log_{10} x \)[/tex]
- This is a logarithmic function because the base is 10, which is greater than 0 and not equal to 1. Therefore, the reason given in this statement is incorrect.
2. Statement 2: [tex]\( y = \log_{\sqrt{3}} x \)[/tex]
- This is also a logarithmic function because the base [tex]\(\sqrt{3}\)[/tex] (which is approximately 1.732) is greater than 0 and not equal to 1. The fact that the base is a square root does not disqualify it from being a logarithmic function. Hence, this statement is incorrect.
3. Statement 3: [tex]\( y = \log_1 x \)[/tex]
- This is not a logarithmic function, and the reason given is correct. The base is 1, and the function [tex]\( y = \log_1 x \)[/tex] is undefined because a base of 1 does not satisfy the conditions for a logarithmic function.
4. Statement 4: [tex]\( y = \log_{\frac{3}{4}} x \)[/tex]
- This is a logarithmic function because the base [tex]\(\frac{3}{4}\)[/tex] is a positive number greater than 0 and not equal to 1. The fact that the base is a fraction does not disqualify it from being a logarithmic function. Therefore, this statement is incorrect.
After evaluating all the statements, we conclude that the true statement is:
Statement 3: [tex]\( y = \log_1 x \)[/tex] is not a logarithmic function because the base is equal to 1.
