Final answer:
A system of equations to represent the equation log4(x+3) = log2(2+x) can be created by expressing both sides with a common logarithm base, leading to equations y = log4(x+3) and y = (1/2) * log4(2+x). This allows for the setting of one expression equal to the other and the solving for x.
Step-by-step explanation:
To find a system of equations that represents the equation log4(x+3) = log2(2+x), we need to use the properties of logarithms. Since the bases of the logarithms are different, we cannot directly equate the arguments of the logarithms (x+3 and 2+x). However, we can apply the property that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers (log xy = log x + log y), to rewrite one of these logarithms in a base that matches the other. In this case, we can express log2 as log4 by using the change of base formula or by expressing 2 as 4^(1/2), since 4 is 2 squared. We can make the base the same and then compare the arguments to find the solution.
One possibility for creating a system of equations could look something like this:
- let y = log4(x+3)
- let y = (1/2) * log4(2+x), because log4(2) is equal to (1/2)
Now we have a system where:
- log4(x+3) = y
- (1/2) * log4(2+x) = y
We equate the two expressions for y to find the value of x:
- log4(x+3) = (1/2) * log4(2+x)
Upon solving this for x, we get to the solution of the original equation.