Use the Definition of a Logarithm to Solve Logarithmic Equations
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$$\log _ { 2 } ( x + 1 ) + \log _ { 2 } ( x - 5 ) = 4$$
A) $\{ 7 \}$
B) $\{ 7 , - 3 \}$
C) $\{ - 3 \}$
D) $\{ 8 \}$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, the equation can be rewritten as $\log_2((x+1)(x-5)) = 4$. Simplifying, this becomes $\log_2(x^2-4x-5) = 4$. Converting to exponential form gives $2^4 = x^2 - 4x - 5$, leading to $x^2 - 4x - 21 = 0$. Factoring or using the quadratic formula yields $x = 7$ or $x = -3$. However, $x = -3$ is not in the domain of the original logarithmic expressions because it would make the argument of at least one of the logarithms negative. Therefore, the correct answer is $x = 7$.

Use the Definition of a Logarithm to Solve Logarithmic Equations log⁡2(x+1)+log⁡2(x−5)=4\log _ { 2 } ( x + 1 ) + \log _ { 2 } ( x - 5 ) = 4log2​(x+1)+log2​(x−5)=4

Use the Definition of a Logarithm to Solve Logarithmic Equations $\log _ { 2 } ( x + 1 ) + \log _ { 2 } ( x - 5 ) = 4$