which properties are present in a table that represents a logarithmic function in the form $y = \\log_{b}x$…
which properties are present in a table that represents a logarithmic function in the form $y = \\log_{b}x$ when $b>1$?\ni. the y - values are always increasing or always decreasing.\nii. the point $(0,1)$ exists in the table.\niii. the y - values will decrease rapidly as the x - values approach zero.\niv. there will only be one x - value in the table with a y - value of zero.\n\no i only\no i and ii only\no i, iii, and iv\no ii and iii only
Answer
Explanation:
Step1: Analyze property I
For $y = \log_b x$ with $b>1$, the function is increasing. So the $y$-values are always increasing as $x$ increases. This property is true.
Step2: Analyze property II
When $x = 0$, $y=\log_b x$ is undefined since the logarithm of 0 is not a real - number. So the point $(0,1)$ does not exist in the table. This property is false.
Step3: Analyze property III
As $x\rightarrow0^{+}$, $y = \log_b x\rightarrow-\infty$. So the $y$-values will decrease rapidly as the $x$-values approach zero. This property is true.
Step4: Analyze property IV
Set $y = 0$, then $\log_b x=0$. By the definition of logarithms, $x = b^0=1$. So there is only one $x$-value (i.e., $x = 1$) for which $y = 0$. This property is true.
Answer:
C. I, III, and IV