Solve the equation.
-$$\log _ { 4 } ( x + 2 ) = 2 + \log _ { 4 } ( x - 4 )$$
A) $\left\{ - \frac { 22 } { 5 } \right\}$
B) $\left\{ - \frac { 2 } { 5 } \right\}$
C) $\left\{ \frac { 2 } { 5 } \right\}$
D) $\left\{ \frac { 22 } { 5 } \right\}$

Question

Quizplus · Accepted Answer

To solve the equation $\log_4(x + 2) = 2 + \log_4(x - 4)$, we can use the properties of logarithms. First, we can subtract $\log_4(x - 4)$ from both sides to get $\log_4(x + 2) - \log_4(x - 4) = 2$. Using the property that $\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})$, we can rewrite the equation as $\log_4\left(\frac{x + 2}{x - 4}\right) = 2$. To remove the logarithm, we can exponentiate both sides with base 4, getting $\frac{x + 2}{x - 4} = 4^2 = 16$. Solving for $x$, we get $x + 2 = 16(x - 4)$, which simplifies to $x + 2 = 16x - 64$. Bringing all terms to one side gives $64 + 2 = 16x - x$, which simplifies to $66 = 15x$, and thus $x = \frac{66}{15} = \frac{22}{5}$. Therefore, the correct answer is $\left\{ \frac{22}{5} \right\}$.

Solve the Equation log⁡4(x+2)=2+log⁡4(x−4)\log _ { 4 } ( x + 2 ) = 2 + \log _ { 4 } ( x - 4 )log4​(x+2)=2+log4​(x−4)

Solve the Equation $\log _ { 4 } ( x + 2 ) = 2 + \log _ { 4 } ( x - 4 )$