Solve the equation. $\log _ { 5 } x = \log _ { 5 } \frac { 1 } { x } + 8$
A) $\quad x = 637$
B) $\quad x = - 625$
C) $\quad x = - 637$
D) $\quad x = 626$
E) $\quad x = - 626$
F) $x = 625$

Question

Quizplus · Accepted Answer

Using the property of logarithms, we can simplify the equation as follows:
\begin{align*}
\log_5 x &= \log_5 \frac{1}{x} + 8\
\log_5 x &= \log_5 1 - \log_5 x + 8\
2\log_5 x &= 9\
\log_5 x &= \frac{9}{2}
\end{align*}
Therefore, $x = 5^{9/2} = 625$.
Note that all the other choices for $x$ are either negative or have absolute value greater than $625$, which would result in a negative value for $\log_5 \frac{1}{x}$, violating the domain of logarithmic functions.

Solve the Equation log⁡5x=log⁡51x+8\log _ { 5 } x = \log _ { 5 } \frac { 1 } { x } + 8log5​x=log5​x1​+8

Solve the Equation $\log _ { 5 } x = \log _ { 5 } \frac { 1 } { x } + 8$