Question

which table represents the graph of a logarithmic function with both an x - and y - intercept?
done

Explanation:

Step1: Recall the domain of logarithmic functions

The domain of a logarithmic function \(y = \log_b(x - h)+k\) is \(x>h\). Logarithmic functions are not defined for non - positive values of the argument of the logarithm (when \(b>0,b
eq1\)). So, we can eliminate the table with \(x=-1.5\) and \(x = - 0.5\) (the second table) since the argument of a basic logarithmic function \(y=\log(x)\) has a domain \(x>0\) and even for a transformed function \(y=\log_b(x - h)+k\), the values of \(x\) for which \(x - h\leq0\) are not in the domain.

Step2: Recall the \(x\) and \(y\) - intercepts

The \(x\) - intercept of a function \(y = f(x)\) is a value of \(x\) such that \(y = 0\). The \(y\) - intercept is a value of \(y\) when \(x = 0\). For the first table:

• The \(x\) - intercept is \(x = 3\) (since \(y = 0\) when \(x = 3\)). But if we assume a general form \(y=\log_b(x - h)+k\), when \(x = 4,y=-15\) is an extreme value. Logarithmic functions \(y=\log_b(x)\) have a relatively slow rate of change.

For the third table:

• We can check the general behavior of a logarithmic function \(y=\log(x)\) which has a vertical asymptote at \(x = 0\). A transformed logarithmic function \(y=\log(x - h)+k\). The third table has values of \(x>0\) (satisfying the domain condition for a basic - type logarithmic function \(y=\log(x)\) or \(y = a\log(x)+b\)).
• We can also check the rate of change. The difference in \(y\) - values:
• When \(x\) changes from \(x_1=0.5\) to \(x_2 = 1.5\) (\(\Delta x=1\)), \(y\) changes from \(y_1=-0.631\) to \(y_2 = 0.369\) (\(\Delta y=0.369-(-0.631)=1\)). When \(x\) changes from \(x_2 = 1.5\) to \(x_3=2.5\) (\(\Delta x = 1\)), \(y\) changes from \(y_2=0.369\) to \(y_3 = 0.834\) (\(\Delta y=0.834 - 0.369=0.465\)). When \(x\) changes from \(x_3=2.5\) to \(x_4=3.5\) (\(\Delta x = 1\)), \(y\) changes from \(y_3=0.834\) to \(y_4 = 1.146\) (\(\Delta y=1.146 - 0.834 = 0.312\)). This is consistent with the slow - growth rate of a logarithmic function \(y=\log(x)\) (after vertical and horizontal transformations).

Answer:

The third table.