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)
The problem requires us to find a point that lies on the graph of f ( x ) = lo g 2 ( x + 3 ) + 2 , given that ( 8 , 3 ) lies on g ( x ) = lo g 2 x .
We test each point by plugging its x-coordinate into f ( x ) and checking if the result matches the y-coordinate.
For the point ( 5 , 5 ) , we have f ( 5 ) = lo g 2 ( 5 + 3 ) + 2 = lo g 2 ( 8 ) + 2 = 3 + 2 = 5 , which matches the y-coordinate.
Therefore, the point ( 5 , 5 ) lies on the graph of f ( x ) . ( 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 . This means that g ( 8 ) = lo g 2 8 = 3 , which is true since 2 3 = 8 . We want to find which of the given points lies on the graph of f ( x ) = lo g 2 ( x + 3 ) + 2 .
Testing the Points We need to test each point ( x , y ) to see if it satisfies the equation y = lo g 2 ( x + 3 ) + 2 .
Testing (5,1) Let's test the point ( 5 , 1 ) . We have 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 (5,5) Now let's test the point ( 5 , 5 ) . We have 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 (11,1) Let's test the point ( 11 , 1 ) . We have f ( 11 ) = lo g 2 ( 11 + 3 ) + 2 = lo g 2 ( 14 ) + 2 . Since lo g 2 ( 14 ) is between 3 and 4, f ( 11 ) is between 5 and 6, so f ( 11 ) = 1 . Thus, the point ( 11 , 1 ) does not lie on the graph of f ( x ) .
Testing (11,5) Finally, let's test the point ( 11 , 5 ) . We have f ( 11 ) = lo g 2 ( 11 + 3 ) + 2 = lo g 2 ( 14 ) + 2 . As we found before, f ( 11 ) is between 5 and 6, so f ( 11 ) = 5 . Thus, the point ( 11 , 5 ) does not lie on the graph of f ( x ) .
Conclusion Therefore, the only point that lies on the graph of f ( x ) = lo g 2 ( x + 3 ) + 2 is ( 5 , 5 ) .
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 of a substance using pH, and modeling population growth or radioactive decay. In finance, logarithmic scales are used to analyze stock market trends and investment growth. Understanding how transformations affect logarithmic graphs helps in interpreting these models and making informed decisions.
