Consider the equation below.
$\log _4(x+3)=\log _2(2+x)$
Which system of equations can represent the equation?
A. $y_1=\frac{\log (x+3)}{\log 4}, y_2=\frac{\log (2+x)}{\log 2}$
B. $y_1=\frac{\log x+3}{\log 4}, y_2=\frac{\log 2+x}{\log 2}$
C. $y_1=\frac{\log 4}{\log 2}, y_2=\frac{\log (x+3)}{\log (2+x)}$
D. $y_1=\frac{\log x+3}{4}, y_2=\frac{\log 2+x}{2}$
Answer (1)
Apply the change of base formula to rewrite lo g 4 ( x + 3 ) as l o g ( 4 ) l o g ( x + 3 ) and define it as y 1 .
Apply the change of base formula to rewrite lo g 2 ( 2 + x ) as l o g ( 2 ) l o g ( 2 + x ) and define it as y 2 .
Form the system of equations: y 1 = l o g ( 4 ) l o g ( x + 3 ) and y 2 = l o g ( 2 ) l o g ( 2 + x ) .
The correct system of equations is: y 1 = lo g 4 lo g ( x + 3 ) , y 2 = lo g 2 lo g ( 2 + x ) .
Explanation
Understanding the Problem We are given the equation lo g 4 ( x + 3 ) = lo g 2 ( 2 + x ) and asked to find an equivalent system of equations. The key idea here is to use the change of base formula for logarithms to express both sides of the equation in terms of a common base, typically the natural logarithm (ln) or the common logarithm (log, base 10).
Change of Base Formula The change of base formula states that lo g a b = l o g c a l o g c b for any positive a , b , c where a = 1 and c = 1 . We will use this formula to rewrite both sides of the given equation using the common logarithm (base 10).
Rewriting the Left Side Applying the change of base formula to the left side of the equation, we have lo g 4 ( x + 3 ) = lo g ( 4 ) lo g ( x + 3 ) . Let's define y 1 = lo g 4 ( x + 3 ) , so we can write y 1 = lo g ( 4 ) lo g ( x + 3 ) .
Rewriting the Right Side Similarly, applying the change of base formula to the right side of the equation, we have lo g 2 ( 2 + x ) = lo g ( 2 ) lo g ( 2 + x ) . Let's define y 2 = lo g 2 ( 2 + x ) , so we can write y 2 = lo g ( 2 ) lo g ( 2 + x ) .
Finding the System of Equations Since lo g 4 ( x + 3 ) = lo g 2 ( 2 + x ) , we have y 1 = y 2 . Thus, the system of equations that represents the given equation is y 1 = lo g ( 4 ) lo g ( x + 3 ) , y 2 = lo g ( 2 ) lo g ( 2 + x ) . Comparing this with the given options, we see that the first option matches our result.
Final Answer Therefore, the system of equations that can represent the given equation is y 1 = lo g 4 lo g ( x + 3 ) , y 2 = lo g 2 lo 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. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to manipulate and solve logarithmic equations is crucial for making informed decisions and predictions in these areas. For example, if you know the initial investment, the interest rate, and the desired final amount, you can use logarithms to find out how long it will take to reach your financial goal.
