Which points lie on the graph of $f(x)=\log _9 x$? Check all that apply.
$\left(-\frac{1}{81}, 2\right)$
$(0,1)$
$\left(\frac{1}{9},-1\right)$
$(3,243)$
$(9,1)$
$(81,2)$
Answer (1)
Check if each point ( x , y ) satisfies the equation 9 y = x .
For ( − 81 1 , 2 ) , 9 2 = 81 = − 81 1 .
For ( 0 , 1 ) , 9 1 = 9 = 0 .
For ( 9 1 , − 1 ) , 9 − 1 = 9 1 .
For ( 3 , 243 ) , 9 243 = 3 .
For ( 9 , 1 ) , 9 1 = 9 .
For ( 81 , 2 ) , 9 2 = 81 .
The points that lie on the graph are ( 9 1 , − 1 ) , ( 9 , 1 ) , ( 81 , 2 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = lo g 9 x . We need to check which of the given points lie on the graph of this function. A point ( x , y ) lies on the graph if and only if f ( x ) = y . In other words, we need to check if lo g 9 x = y for each point. This is equivalent to checking if 9 y = x .
Checking Each Point Let's check each point:
( − 81 1 , 2 ) : We need to check if 9 2 = − 81 1 . Since 9 2 = 81 , this point does not lie on the graph.
( 0 , 1 ) : We need to check if 9 1 = 0 . Since 9 1 = 9 , this point does not lie on the graph. Also, note that the logarithm function is not defined at x = 0 .
( 9 1 , − 1 ) : We need to check if 9 − 1 = 9 1 . Since 9 − 1 = 9 1 , this point lies on the graph.
( 3 , 243 ) : We need to check if 9 243 = 3 . Since 9 243 is a very large number and not equal to 3, this point does not lie on the graph.
( 9 , 1 ) : We need to check if 9 1 = 9 . Since 9 1 = 9 , this point lies on the graph.
( 81 , 2 ) : We need to check if 9 2 = 81 . Since 9 2 = 81 , this point lies on the graph.
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 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 solution, and modeling population growth or radioactive decay. In finance, logarithmic scales are used to analyze stock market trends and investment growth. Understanding how to verify points on a logarithmic graph helps in interpreting these applications and making informed decisions based on the data.
