Question

question 8 · 1 point
graph the logarithmic function below by moving the key point.
( y = log_4(x) )
you can only graph integer points, for example you cannot plot the point ( left( -1, \frac{1}{4}
ight) ).
provide your answer below:

Explanation:

Step1: Recall log function key point

For \( y = \log_b(x) \), when \( x = 1 \), \( y=\log_b(1)=0 \); when \( x = b \), \( y=\log_b(b)=1 \).
For \( y = \log_4(x) \), at \( x = 1 \), \( y = \log_4(1)=0 \) (point \((1,0)\)); at \( x = 4 \), \( y=\log_4(4)=1 \) (point \((4,1)\)).

Step2: Identify integer points

We need integer - valued \( x \) and \( y \). Check \( x = 1 \): \( y = 0 \) (valid, integer point). Check \( x = 4 \): \( y = 1 \) (valid, integer point). Also, \( x=\frac{1}{4} \) gives \( y=\log_4(\frac{1}{4})=- 1 \) (but \( x=\frac{1}{4} \) is not an integer - valued \( x \) in terms of the grid's integer - valued \( x \) positions we can plot? Wait, the problem says "you can only graph integer points" (probably integer \( x \) and \( y \)). So \( x = 1,y = 0 \) and \( x = 4,y = 1 \) and \( x=\frac{1}{4},y=- 1 \) (but \( x=\frac{1}{4} \) is not integer). The key point we can move is, for example, the point \((1,0)\) (since when \( x = 1 \), \( y = 0 \) is an integer point) or \((4,1)\). The most basic key point for a logarithmic function \( y=\log_b(x) \) is \((1,0)\) (the \( x \) - intercept) and \((b,1)\). Since we need to move the key point, the point \((1,0)\) is a key point (as \( \log_b(1)=0 \) for any \( b>0,b
eq1 \)) and \((4,1)\) is also a key point. But if we consider the standard key point to plot first, \((1,0)\) is a key point (the \( x \) - intercept) and \((4,1)\) is another.

To graph \( y=\log_4(x) \), we can plot the point \((1,0)\) (since at \( x = 1 \), \( y = 0 \), both integers) and \((4,1)\) (at \( x = 4 \), \( y = 1 \), both integers). The point \((1,0)\) is a fundamental key point for logarithmic functions (the intersection with the \( x \) - axis).

Answer:

The key points we can plot are \((1,0)\) and \((4,1)\). If we move the key point, the point \((1,0)\) (where \( x = 1,y = 0 \)) is a key point for the function \( y=\log_4(x) \) (and also \((4,1)\)). So we can plot the point \((1,0)\) (or \((4,1)\)) on the graph. The main key point (the \( x \) - intercept) is \((1,0)\).