Given that the point $(8,3)$ lies on the graph of $g(x)=\log _2 x$, which point lies on the graph of $f(x)=\log _2(x+3)+2$?
A. $(5,1)$
B. $(5,5)$
C. $(11,1)$
D. $(11,5)$
Answer (1)
Substitute each given point into the equation f ( x ) = lo g 2 ( x + 3 ) + 2 .
For the point (5,1): f ( 5 ) = lo g 2 ( 5 + 3 ) + 2 = lo g 2 ( 8 ) + 2 = 3 + 2 = 5 = 1 .
For the point (5,5): f ( 5 ) = lo g 2 ( 5 + 3 ) + 2 = lo g 2 ( 8 ) + 2 = 3 + 2 = 5 .
Therefore, the point (5,5) lies on the graph of f ( x ) .
The answer is ( 5 , 5 ) .
Explanation
Understanding the Problem We are given that the point ( 8 , 3 ) lies on the graph of g ( x ) = lo g 2 x . We want to find which of the given points lies on the graph of f ( x ) = lo g 2 ( x + 3 ) + 2 . This means we need to check each point ( x , y ) to see if it satisfies the equation y = lo g 2 ( x + 3 ) + 2 .
Testing Point (5,1) Let's test the point ( 5 , 1 ) . We plug in x = 5 into f ( x ) to get f ( 5 ) = lo g 2 ( 5 + 3 ) + 2 = lo g 2 ( 8 ) + 2 = 3 + 2 = 5 . Since f ( 5 ) = 5 = 1 , the point ( 5 , 1 ) does not lie on the graph of f ( x ) .
Testing Point (5,5) Now let's test the point ( 5 , 5 ) . We plug in x = 5 into f ( x ) to get f ( 5 ) = lo g 2 ( 5 + 3 ) + 2 = lo g 2 ( 8 ) + 2 = 3 + 2 = 5 . Since f ( 5 ) = 5 , the point ( 5 , 5 ) lies on the graph of f ( x ) .
Testing Point (11,1) Let's test the point ( 11 , 1 ) . We plug in x = 11 into f ( x ) to get f ( 11 ) = lo g 2 ( 11 + 3 ) + 2 = lo g 2 ( 14 ) + 2 . Since 2 3 = 8 < 14 < 16 = 2 4 , we have 3 < lo g 2 ( 14 ) < 4 . Therefore, 5 < lo g 2 ( 14 ) + 2 < 6 . Since f ( 11 ) = lo g 2 ( 14 ) + 2 = 1 , the point ( 11 , 1 ) does not lie on the graph of f ( x ) .
Testing Point (11,5) Finally, let's test the point ( 11 , 5 ) . We plug in x = 11 into f ( x ) to get f ( 11 ) = lo g 2 ( 11 + 3 ) + 2 = lo g 2 ( 14 ) + 2 . As we found before, 5 < lo g 2 ( 14 ) + 2 < 6 . Therefore, f ( 11 ) = lo g 2 ( 14 ) + 2 = 5 , so the point ( 11 , 5 ) does not lie on the graph of f ( x ) .
Conclusion Therefore, the point ( 5 , 5 ) lies on the graph of f ( x ) = lo g 2 ( x + 3 ) + 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. Understanding how transformations affect logarithmic graphs, as in this problem, helps in adjusting these models to fit different scenarios. For example, if we are modeling the spread of a disease, adding a constant to the input variable (x+3) might represent a time delay in the onset of symptoms, while adding a constant to the function (+2) could represent an increase in the rate of infection.
