Solve the equation. $\log _ { 3 } x = \log _ { 3 } \frac { 1 } { x } + 6$
A) $\quad x = 30$
B) $\quad x = 16$
C) $\quad x = - 27$
D) $x = 27$
E) $\quad x = - 16$
F) $\quad x = - 30$

Question

Quizplus · Accepted Answer

To solve the equation $\log_3 x = \log_3 \frac{1}{x} + 6$, we can use the properties of logarithms. First, recognize that $\log_3 \frac{1}{x}$ is equivalent to $-\log_3 x$ because $\log_3 x^{-1} = -\log_3 x$. This gives us the equation $\log_3 x = -\log_3 x + 6$. Adding $\log_3 x$ to both sides to isolate the 6 gives $2\log_3 x = 6$, which simplifies to $\log_3 x = 3$. Converting back from logarithmic to exponential form, we get $3^3 = x$, which means $x = 27$.

Solve the Equation log⁡3x=log⁡31x+6\log _ { 3 } x = \log _ { 3 } \frac { 1 } { x } + 6log3​x=log3​x1​+6

Solve the Equation $\log _ { 3 } x = \log _ { 3 } \frac { 1 } { x } + 6$