Solve the equation and express the solution in exact form.
-$\log _ { 4 } ( x + 1 ) + \log _ { 4 } ( x - 1 ) = 2$
A) $\left\{ \frac { 3 } { 2 } \right\}$
B) $\{ 17 \}$
C) $\{ \sqrt { 17 } \}$
D) $\{ 8 \}$

Question

Quizplus · Accepted Answer

Using the property of logarithms that allows combining two logs with the same base into a single log (logarithm of a product), the equation becomes $\log_4((x+1)(x-1)) = 2$. Simplifying inside the log gives $\log_4(x^2-1) = 2$. To solve for $x$, convert the logarithmic equation to its exponential form: $4^2 = x^2 - 1$, which simplifies to $16 = x^2 - 1$. Solving for $x^2$ gives $x^2 = 17$, and thus $x = \sqrt{17}$.

Solve the Equation and Express the Solution in Exact Form log⁡4(x+1)+log⁡4(x−1)=2\log _ { 4 } ( x + 1 ) + \log _ { 4 } ( x - 1 ) = 2log4​(x+1)+log4​(x−1)=2

Solve the Equation and Express the Solution in Exact Form $\log _ { 4 } ( x + 1 ) + \log _ { 4 } ( x - 1 ) = 2$