Question

which properties are present in a table that represents a logarithmic function in the form ( y = log_{b}x ) when ( b > 1 )?
i. the ( y )-values are always increasing or always decreasing.
ii. the point ( (0, 1) ) exists in the table.
iii. the ( y )-values will decrease rapidly as the ( x )-values approach zero.
iv. there will only be one ( x )-value in the table with a ( y )-value of zero.
o i only
o i and ii only
o i, iii, and iv
o ii and iii only

Explanation:

• For property I:
• The logarithmic function \(y = \log_{b}x\) with \(b>1\) is a one - to - one function. Its derivative is \(y^\prime=\frac{1}{x\ln b}\), and since \(x>0\) (domain of \(y = \log_{b}x\)) and \(b > 1\) (so \(\ln b>0\)), the function is increasing for all \(x>0\). So the \(y\) - values are always increasing.
• For property II:
• Substitute \(x = 0\) into \(y=\log_{b}x\). The function \(y = \log_{b}x\) is not defined at \(x = 0\) because \(\lim_{x

ightarrow0^{+}}\log_{b}x=-\infty\). The point \((0,1)\) does not lie on the graph of \(y=\log_{b}x\).

• For property III:
• As \(x

ightarrow0^{+}\), using the limit \(\lim_{x
ightarrow0^{+}}\log_{b}x=-\infty\) (since \(b > 1\)). So the \(y\) - values will decrease rapidly as \(x\) - values approach zero.

• For property IV:
• Set \(y = 0\), then \(0=\log_{b}x\). By the definition of logarithms (\(y=\log_{b}x\) if and only if \(x = b^{y}\)), when \(y = 0\), \(x=b^{0}=1\). So there is only one \(x\) (i.e., \(x = 1\)) for which \(y = 0\) on the graph of \(y=\log_{b}x\).

Answer:

C. I, III, and IV