Which points lie on the graph of [tex]f(x)=\log _9 x[/tex]? Check all that apply.
[tex]\left(-\frac{1}{81}, 2\right)[/tex]
[tex](0,1)[/tex]
[tex]\left(\frac{1}{9},-1\right)[/tex]
[tex](3,243)[/tex]
[tex](9,1)[/tex]
[tex](81,2)[/tex]
Answer (1)
Check each point (x, y) to see if it satisfies the equation 9 y = x .
For ( − 81 1 , 2 ) , 9 2 = 81 = − 81 1 , so it's not on the graph.
For ( 0 , 1 ) , 9 1 = 9 = 0 , so it's not on the graph.
For ( 9 1 , − 1 ) , 9 − 1 = 9 1 , so it's on the graph.
For ( 3 , 243 ) , 9 243 = 3 , so it's not on the graph.
For ( 9 , 1 ) , 9 1 = 9 , so it's on the graph.
For ( 81 , 2 ) , 9 2 = 81 , so it's on the graph.
The points on the graph are: ( 9 1 , − 1 ) , ( 9 , 1 ) , ( 81 , 2 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = \t lo g 9 x and a set of points. Our goal is to determine which of these points lie on the graph of the function. A point ( x , y ) lies on the graph if and only if f ( x ) = y , which means lo g 9 x = y . This is equivalent to checking if 9 y = x . We will check each point to see if it satisfies this condition.
Checking the point (-1/81, 2) Let's check the point ( − 81 1 , 2 ) . We need to verify if 9 2 = − 81 1 . Since 9 2 = 81 , this is false.
Checking the point (0, 1) Next, we check the point ( 0 , 1 ) . We need to verify if 9 1 = 0 . Since 9 1 = 9 , this is false.
Checking the point (1/9, -1) Now, we check the point ( 9 1 , − 1 ) . We need to verify if 9 − 1 = 9 1 . Since 9 − 1 = 9 1 , this is true.
Checking the point (3, 243) Let's check the point ( 3 , 243 ) . We need to verify if 9 243 = 3 . Since 9 243 is a very large number and not equal to 3, this is false.
Checking the point (9, 1) Now, we check the point ( 9 , 1 ) . We need to verify if 9 1 = 9 . Since 9 1 = 9 , this is true.
Checking the point (81, 2) Finally, we check the point ( 81 , 2 ) . We need to verify if 9 2 = 81 . Since 9 2 = 81 , this is true.
Final Answer Therefore, the points that lie on the graph of f ( x ) = lo g 9 x are ( 9 1 , − 1 ) , ( 9 , 1 ) , and ( 81 , 2 ) .
Examples
Logarithmic functions are incredibly useful in many real-world scenarios. For example, the Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. An earthquake of magnitude 7 is ten times more powerful than an earthquake of magnitude 6. Similarly, the decibel scale, used to measure sound intensity, is also logarithmic. These scales allow us to represent a wide range of values in a more manageable way, making it easier to understand and compare vastly different quantities.
