which statement is true?\n- $y = \\log_{10} x$ is not a logarithmic function because the base is greater…
which statement is true?\n- $y = \\log_{10} x$ is not a logarithmic function because the base is greater than 0.\n- $y = \\log_{\\sqrt{3}} x$ is not a logarithmic function because the base is a square root.\n- $y = \\log_{1} x$ is not a logarithmic function because the base is equal to 1.\n- $y = \\log_{\\frac{3}{4}} x$ is not a logarithmic function because the base is a fraction.
Answer
Brief Explanations:
To determine the true statement, we recall the definition of a logarithmic function. A logarithmic function is of the form ( y = \log_b x ), where ( b>0 ), ( b\neq1 ), and ( x > 0 ).
- For ( y=\log_{10}x ): The base ( 10>0 ) and ( 10\neq1 ), so it is a logarithmic function. The statement claiming it is not is false.
- For ( y = \log_{\sqrt{3}}x ): The base ( \sqrt{3}\approx1.732>0 ) and ( \sqrt{3}\neq1 ), so it is a logarithmic function. The statement claiming it is not is false.
- For ( y=\log_{1}x ): The base ( b = 1 ). By the definition of a logarithmic function, ( b) must not equal 1 (since ( \log_{1}x=\frac{\ln x}{\ln 1} ), and ( \ln 1 = 0 ), which is undefined). So this is not a logarithmic function, and the statement is true.
- For ( y=\log_{\frac{3}{4}}x ): The base ( \frac{3}{4}>0 ) and ( \frac{3}{4}\neq1 ), so it is a logarithmic function. The statement claiming it is not is false.
Answer:
( y = \log_{1}x ) is not a logarithmic function because the base is equal to 1.