Complete the table for the function [tex]y=\log _{\frac{1}{2}}(x)[/tex].
[tex]
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline$\frac{1}{4}$ & $\square$ \\
\hline$\frac{1}{2}$ & $\square$ \\
\hline 1 & $\square$ \\
\hline 2 & $\square$ \\
\hline 4 & $\square$ \\
\hline
\end{tabular}
[/tex]
Graph the function [tex]y=\log _{\frac{3}{4}}(x)[/tex]. Plot two points with integer coordinates to graph the function.
Answer (2)
The completed table for the function y = lo g 2 1 ( x ) shows values for multiple x inputs, such as ( 4 1 , 2 ) and ( 1 , 0 ) . For the function y = lo g 4 3 ( x ) , one point with integer coordinates is ( 1 , 0 ) , while estimating other non-integer points for graphing. Graphing demonstrates the behavior of the logarithmic function which typically decreases as x increases when the base is less than 1.
;
Complete the table for y = lo g 2 1 ( x ) by evaluating the function at given x values.
Find that the completed table is: x = 4 1 , y = 2 ; x = 2 1 , y = 1 ; x = 1 , y = 0 ; x = 2 , y = − 1 ; x = 4 , y = − 2 .
Identify one point with integer coordinates for y = lo g 4 3 ( x ) as ( 1 , 0 ) .
Note that finding another point with integer coordinates is difficult, suggesting a possible error in the problem statement, but proceed with (1,0) as one point and approximate the graph.
The completed table is:
x
y
4 1
2
2 1
1
1
0
2
− 1
4
− 2
One point with integer coordinates is ( 1 , 0 ) .
( 1 , 0 )
Explanation
Understanding the Problem We are given the function y = lo g 2 1 ( x ) and asked to complete a table of values. We are also asked to graph the function y = lo g 4 3 ( x ) and plot two points with integer coordinates.
Completing the Table To complete the table for y = lo g 2 1 ( x ) , we need to evaluate the function at x = 4 1 , 2 1 , 1 , 2 , 4 . Recall that y = lo g b ( x ) means b y = x .
Calculating y for x=1/4 For x = 4 1 , we have y = lo g 2 1 ( 4 1 ) . Since ( 2 1 ) 2 = 4 1 , we have y = 2 .
Calculating y for x=1/2 For x = 2 1 , we have y = lo g 2 1 ( 2 1 ) . Since ( 2 1 ) 1 = 2 1 , we have y = 1 .
Calculating y for x=1 For x = 1 , we have y = lo g 2 1 ( 1 ) . Since ( 2 1 ) 0 = 1 , we have y = 0 .
Calculating y for x=2 For x = 2 , we have y = lo g 2 1 ( 2 ) . Since ( 2 1 ) − 1 = 2 , we have y = − 1 .
Calculating y for x=4 For x = 4 , we have y = lo g 2 1 ( 4 ) . Since ( 2 1 ) − 2 = 4 , we have y = − 2 .
Completed Table The completed table is:
x
y
4 1
2
2 1
1
1
0
2
− 1
4
− 2
Finding Points for Graphing Now, let's consider the function y = lo g 4 3 ( x ) . We want to find two points with integer coordinates on the graph of this function. We know that for any base b , lo g b ( 1 ) = 0 . Therefore, the point ( 1 , 0 ) is on the graph.
Searching for Integer Coordinates To find another point with integer coordinates, we need to find an x such that lo g 4 3 ( x ) is an integer. Let's try to find an x such that y = 1 . Then x = ( 4 3 ) 1 = 4 3 , which is not an integer. Let's try y = − 1 . Then x = ( 4 3 ) − 1 = 3 4 , which is also not an integer.
Approximating the Graph It seems difficult to find another point with integer coordinates. However, the problem asks us to plot two points with integer coordinates to graph the function. Since we already found the point ( 1 , 0 ) , we can choose another point where x is an integer and approximate the y value. For example, let x = 2 . Then y = lo g 4 3 ( 2 ) = l n 4 3 l n 2 ≈ − 2.41 . So the point is approximately ( 2 , − 2.41 ) . Since we need integer coordinates, the problem might have an error, or it expects us to only plot (1, 0) and approximate the graph.
Final Considerations Since the problem explicitly asks for two points with integer coordinates, and it is difficult to find another such point besides (1, 0), we will proceed with (1,0) as one point. For the second point, we can consider the point where x = 0 , but the logarithm is not defined at x = 0 . Therefore, the problem might have an error.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a chemical solution, and modeling population growth or radioactive decay. Understanding how to evaluate and graph logarithmic functions helps in analyzing and interpreting data in these fields. For example, in seismology, the magnitude of an earthquake is calculated using a logarithmic scale, where each whole number increase represents a tenfold increase in amplitude. By understanding the properties of logarithms, scientists can accurately assess and communicate the severity of seismic events.
