Solve the equation and express the solution in exact form.
-$$\log _ { 3 } x = \sqrt { \log 3 x }$$
A) $\{ 0,1 \}$
B) $\{ 1,3 \}$
C) $\{ 3 \}$
D) $\{ 0,3 \}$

Question

Quizplus · Accepted Answer

The equation can be solved by recognizing that both sides of the equation involve logarithms that can be related through properties of logarithms and exponents. Let's denote $y = \log_3 x$. The equation becomes $y = \sqrt{\log_3 x}$. Squaring both sides gives $y^2 = \log_3 x$. Since $y = \log_3 x$, we have $y^2 = y$, leading to $y^2 - y = 0$, which simplifies to $y(y - 1) = 0$. Thus, $y = 0$ or $y = 1$, which means $\log_3 x = 0$ or $\log_3 x = 1$. Solving these gives $x = 3^0 = 1$ or $x = 3^1 = 3$, so the solution set is $\{1, 3\}$.

Solve the Equation and Express the Solution in Exact Form log⁡3x=log⁡3x\log _ { 3 } x = \sqrt { \log 3 x }log3​x=log3x​

Solve the Equation and Express the Solution in Exact Form $\log _ { 3 } x = \sqrt { \log 3 x }$