Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4
decimal places if necessary.
-$$\log _ { 4 } x - \log _ { 4 } ( 4 x - 7 ) = \frac { 1 } { 2 } \log _ { 4 } 4$$
A) $\{ 2 , - 2 \}$
B) $\{ 2 \}$
C) { }
D) $\{ - 2 \}$

Question

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4
decimal places if necessary.
-$$\log _ { 4 } x - \log _ { 4 } ( 4 x - 7 ) = \frac { 1 } { 2 } \log _ { 4 } 4$$
A) $\{ 2 , - 2 \}$
B) $\{ 2 \}$
C) {  }
D) $\{ - 2 \}$

Quizplus · Accepted Answer

First, simplify the equation using logarithmic properties. The right side of the equation simplifies to $1/2$, since $\log_4 4 = 1$. Then, use the property $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$ to combine the logarithms on the left side:$$\log_4 \left(\frac{x}{4x - 7}\right) = \frac{1}{2}$$Raise both sides as powers of 4 to eliminate the logarithm:$$4^{\log_4 \left(\frac{x}{4x - 7}\right)} = 4^{\frac{1}{2}}$$This simplifies to:$$\frac{x}{4x - 7} = \sqrt{4} = 2$$Solving for $x$:$$x = 2(4x - 7)$$$$x = 8x - 14$$$$7x = 14$$$$x = 2$$Thus, the solution set is $\{2\}$, making choice B correct.

Solve the Equation log⁡4x−log⁡4(4x−7)=12log⁡44\log _ { 4 } x - \log _ { 4 } ( 4 x - 7 ) = \frac { 1 } { 2 } \log _ { 4 } 4log4​x−log4​(4x−7)=21​log4​4

Solve the Equation $\log _ { 4 } x - \log _ { 4 } ( 4 x - 7 ) = \frac { 1 } { 2 } \log _ { 4 } 4$