Consider the equation below.
[tex]$\log _4(x+3)=\log _2(2+x)$[/tex]
Which system of equations can represent the equation
A. [tex]$y_1=\frac{\log (x+3)}{\log 4}, y_2=\frac{\log (2+x)}{\log 2}$[/tex]
B. [tex]$y_1=\frac{\log x+3}{\log 4}, y_2=\frac{\log 2+x}{\log 2}$[/tex]
C. [tex]$y_1=\frac{\log 4}{\log 2}, y_2=\frac{\log (x+3)}{\log (2+x)}$[/tex]
D. [tex]$y_1=\frac{\log x+3}{4}, y_2=\frac{\log 2+x}{2}$[/tex]
Answer (2)
The problem requires finding a system of equations equivalent to a given logarithmic equation. By applying the change of base formula and setting each side of the equation equal to y 1 and y 2 respectively, we express the original equation as a system of two equations. The solution is: y 1 = l o g 4 l o g ( x + 3 ) , y 2 = l o g 2 l o g ( 2 + x ) .
Explanation
Understanding the Problem We are given the equation lo g 4 ( x + 3 ) = lo g 2 ( 2 + x ) . We need to find the system of equations that represents the given equation. The change of base formula is lo g a b = l o g c a l o g c b
Applying Change of Base Formula Apply the change of base formula to both sides of the equation lo g 4 ( x + 3 ) = lo g 2 ( 2 + x ) . Using the change of base formula, rewrite lo g 4 ( x + 3 ) as l o g 4 l o g ( x + 3 ) . Similarly, rewrite lo g 2 ( 2 + x ) as l o g 2 l o g ( 2 + x ) . The equation becomes l o g 4 l o g ( x + 3 ) = l o g 2 l o g ( 2 + x ) .
Defining the System of Equations Define y 1 = l o g 4 l o g ( x + 3 ) and y 2 = l o g 2 l o g ( 2 + x ) . The system of equations is y 1 = l o g 4 l o g ( x + 3 ) , y 2 = l o g 2 l o g ( 2 + x ) .
Final Answer The system of equations that represents the given equation is y 1 = l o g 4 l o g ( x + 3 ) , y 2 = l o g 2 l o g ( 2 + x ) .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding how to manipulate and represent these equations as systems can help in solving complex problems in these areas. For instance, in seismology, the Richter scale uses logarithms to measure the amplitude of seismic waves, and converting logarithmic scales to linear scales is crucial for understanding the actual energy released during an earthquake.
The correct system of equations representing the logarithmic equation lo g 4 ( x + 3 ) = lo g 2 ( 2 + x ) is option A: y 1 = l o g 4 l o g ( x + 3 ) , y 2 = l o g 2 l o g ( 2 + x ) .
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