Question

Find the potential solution to the equation log₄(2 – x) = log₄(–5x – 18). x = -5 which statement is true about the potential solution x = –5?

a. this number is not a true solution because it is negative.
b. this number is not a true solution because –5x – 18 is negative.
c. this number is not a true solution because 2 – x is negative.
d. this number is a true solution of the original equation.

Answer 1

After substituting x with -5 in the equation log₄(2 – x) = log₄(–5x – 18), both expressions inside the logs equal 7, which is positive. Therefore, x = -5 is a true solution to the original equation.Option D is the correct answer.

Step-by-step explanation:

To determine whether x = -5 is a solution to the equation log₄(2 – x) = log₄(–5x – 18), we substitute x with -5 in both expressions within the logarithms:

1. log₄(2 – (-5)) = log₄(2 + 5) = log₄(7), which is defined because 7 is positive.
2. log₄(–5(-5) – 18) = log₄(25 – 18) = log₄(7), which is also defined because 7 is positive.

Given that both expressions within the logarithms are equal and positive, the logs are equal, and therefore, x = -5 satisfies the original equation. In logarithmic equations, it is essential that the arguments of the logarithms (the numbers inside the log function) be positive, otherwise the logarithm is undefined.

There is no restriction that x itself cannot be negative; what matters is that the arguments of the logarithms are positive after substituting x.

Thus, the correct statement is:

d. This number is a true solution of the original equation.