The solution of the equation $3^{\log_a x}+3x^{\log_a 3}=2$

Question

The solution of the equation $$3^{\log_a x}+3x^{\log_a 3}=2$$ I tried to solve by taking logarithm again and bringing the power to the same level and changing two given in the R.H.S to $$2\log_3 3$$$but could not get the result please help me

Answer 1

Hint:

Let $\log_ax=y\implies a^y=x$

$$2=3^y+3(a^y)^{\log_a3}=3^y+3(a^{\log_a3})^y=3^y+3(3)^y$$

Hope this is sufficient!

Answer 2

Hint: $$3^{\log_{a}{x}}+3x^{\log_{a}{x}}=4\cdot 3^{\log_{a}{x}}$$ $$x=2^{\log_{3}{a}}$$

Answer 3

$$x^{\log_a 3}=3^{\log_a x}, where \hspace{5pt} x>0 $$ Assume $$3^{\log_a x}=t$$

$$t+3 \cdot t=2$$