Solve the equation.
-$\log _ { 5 } x + \log _ { 5 } ( x - 24 ) = 2$
A) $\{ 25 \}$
B) $\{ 1 , - 25 \}$
C) $\{ - 1,25 \}$
D) $\{ 5 \}$

Question

Quizplus · Accepted Answer

Using the logarithm property $\log_b(m) + \log_b(n) = \log_b(mn)$, the equation becomes $\log_5(x(x - 24)) = 2$. Solving for $x$, we get $x^2 - 24x = 5^2$, leading to $x^2 - 24x - 25 = 0$. Factoring gives $(x - 25)(x + 1) = 0$, so $x = 25$ or $x = -1$. However, $x = -1$ is not valid because it would result in a negative argument in the original logarithmic equation. Therefore, the only solution is $x = 25$.

Solve the Equation log⁡5x+log⁡5(x−24)=2\log _ { 5 } x + \log _ { 5 } ( x - 24 ) = 2log5​x+log5​(x−24)=2

Solve the Equation $\log _ { 5 } x + \log _ { 5 } ( x - 24 ) = 2$