Jerome solved the equation below by graphing.
[tex]\log _2 x+\log _2(x-2)=3[/tex]
Which of the following shows the correct system of equations and solution?
A. [tex]$y_1=\frac{\log x}{\log 2}+\frac{\log (x-2)}{\log 2}, y_2=3 ; x=3$[/tex]
B. [tex]$y_1=\frac{\log x}{\log 2}+\frac{\log (x-2)}{\log 2}, y_2=3 ; x=4$[/tex]
C. [tex]$y_1=\log x+\log (x-2), y_2=3 ; x=33$[/tex]
D. [tex]$y_1=\log x+\log (x-2), y_2=3 ; x=44$[/tex]
Answer (1)
Rewrite the logarithmic equation as a system of equations: y 1 = lo g 2 x + lo g 2 ( x − 2 ) and y 2 = 3 .
Use the change of base formula to express y 1 in terms of common logarithms: y 1 = l o g 2 l o g x + l o g 2 l o g ( x − 2 ) .
Substitute x = 3 and x = 4 into the original equation to check for validity.
Conclude that the correct system of equations and solution is y 1 = l o g 2 l o g x + l o g 2 l o g ( x − 2 ) , y 2 = 3 ; x = 4 , since x = 4 satisfies the original equation. y 1 = lo g 2 lo g x + lo g 2 lo g ( x − 2 ) , y 2 = 3 ; x = 4
Explanation
Problem Analysis The problem requires us to identify the correct system of equations and the solution to the given logarithmic equation: lo g 2 x + lo g 2 ( x − 2 ) = 3 . We will analyze each option to determine the correct system and solution.
System of Equations First, let's rewrite the given equation as a system of two equations: y 1 = lo g 2 x + lo g 2 ( x − 2 ) and y 2 = 3 . To compare with the options, we can use the change of base formula to rewrite y 1 in terms of common logarithms (base 10): y 1 = l o g 2 l o g x + l o g 2 l o g ( x − 2 ) .
Checking the Options Now, let's examine the given options:
Option 1: y 1 = l o g 2 l o g x + l o g 2 l o g ( x − 2 ) , y 2 = 3 ; x = 3 Option 2: y 1 = l o g 2 l o g x + l o g 2 l o g ( x − 2 ) , y 2 = 3 ; x = 4 Option 3: y 1 = lo g x + lo g ( x − 2 ) , y 2 = 3 ; x = 33 Option 4: y 1 = lo g x + lo g ( x − 2 ) , y 2 = 3 ; x = 44
Options 1 and 2 have the correct system of equations using the change of base formula. Options 3 and 4 do not use the change of base formula correctly.
Verifying the Solution Next, we need to check which of the solutions, x = 3 or x = 4 , is correct. We substitute each value into the original equation lo g 2 x + lo g 2 ( x − 2 ) = 3 .
If x = 3 , then lo g 2 3 + lo g 2 ( 3 − 2 ) = lo g 2 3 + lo g 2 1 = lo g 2 3 + 0 = lo g 2 3 . Since lo g 2 3 ≈ 1.585 , this is not equal to 3. Therefore, x = 3 is not the correct solution.
If x = 4 , then lo g 2 4 + lo g 2 ( 4 − 2 ) = lo g 2 4 + lo g 2 2 = 2 + 1 = 3 . Therefore, x = 4 is the correct solution.
Conclusion Based on our analysis, the correct system of equations and solution is: y 1 = l o g 2 l o g x + l o g 2 l o g ( x − 2 ) , y 2 = 3 ; x = 4 .
Examples
Logarithmic equations are used in various fields such as computer science, finance, and physics. For example, in computer science, logarithms are used to analyze the time complexity of algorithms. In finance, they are used to calculate the time it takes for an investment to double at a given interest rate. Understanding how to solve logarithmic equations is essential for solving real-world problems in these fields. The equation lo g 2 x + lo g 2 ( x − 2 ) = 3 can be seen as a simplified model for calculating the growth rate or time required in certain scenarios.
