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 the logarithms with a common base.
Express both sides of the original equation as separate equations y 1 and y 2 .
Identify the system of equations that matches the derived expressions.
The correct 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 ) .
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 from the given options. To do this, we can use the change of base formula to express both logarithms in terms of a common base. 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 Let's use the common logarithm (base 10) to rewrite both sides of the equation. Using the change of base formula, we have: lo g 4 ( x + 3 ) = lo g 4 lo g ( x + 3 ) lo g 2 ( 2 + x ) = lo g 2 lo g ( 2 + x )
Creating the System of Equations Now, we can set up a system of equations where y 1 represents the left side of the original equation and y 2 represents the right side: y 1 = lo g 4 lo g ( x + 3 ) y 2 = lo g 2 lo g ( 2 + x )
Identifying the Correct Option Comparing this system of equations with the given options, we find that the first option matches our result: 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. In finance, logarithmic scales are used to analyze stock market trends and investment growth. For instance, comparing investment growth using logarithmic scales helps visualize percentage changes more clearly, especially when dealing with large numbers or exponential growth.
