Final answer:
The extraneous solution to the logarithmic equation is log₄ (-7x + 21). Option c is correct.
Step-by-step explanation:
To identify the extraneous solution to the logarithmic equation, we need to solve each of the given logarithmic equations and check for any solutions that do not satisfy the original equation.
The given logarithmic equations are: a. log₄ (x) b. log₄ (x - 3) c. log₄ (-7x + 21)
We will solve each equation and then determine which one yields an extraneous solution.
a. log₄ (x): The domain of a logarithmic function is restricted to positive real numbers, as the argument of the logarithm must be greater than 0.
Therefore, for log₄ (x), x must be greater than 0.
b. log₄ (x - 3): Similar to the first equation, the domain of this logarithmic function is also restricted to positive real numbers.
Thus, x - 3 must be greater than 0, leading to x > 3.
c. log₄ (-7x + 21): In this case, we need to find the values of x that make -7x + 21 greater than 0, as per the domain of a logarithmic function.
Solving for x, we get: -7x + 21 > 0
-7x > -21
x < 3
Comparing the solutions obtained for each equation with their respective domains, we find that for equation c (log₄ (-7x + 21)), the solution x < 3 violates the domain requirement for a logarithmic function.
Therefore, x < 3 is an extraneous solution for this equation.
Extraneous Solution: log₄ (-7x + 21) -> x < 3
Therefore, the extraneous solution to the logarithmic equation is log₄ (-7x + 21). Option c is correct.