I made a mistake copying the question ! it should be $x^{\log_3{(x)}-1} = 3^{(3+\log_3{x})}$.
I'm trying to solve the equation: $x^{\log_3{(x-1)}} = 3^{(3+\log_3{x})}$
What I tried:
I took the logarithm base 3 of both sides which resulted in: $\log_3{x^{\log_3{(x-1)}}} = 3 + \log_3{x}$
I then used the logarithm property $\log_b(a^c) = c \log_b(a)$ to get: $\log_3{x} \cdot \log_3{(x-1)} = 3 + \log_3{x}$
I tried to express the right-hand side as a product, realizing that: $3^{(3+\log_3{x})}$ could be expressed as: $3^3 \times 3^{\log_3{x}}$ which becomes: 27x
However, I am struggling to find a way to simplify or solve the equation from here.
The asnwers should be 27, $1/3$
Question: How can I solve this equation either algebraically or using any special methods?
Any guidance or assistance is greatly appreciated!