## SUBJECT II

Solve the following logarithmic equation: $x^{\log _{3}(x-1)}+2 \cdot(x-1)^{\log _{3} x}=3 \cdot x^{2}$
## SUBJECT II

Solve the following logarithmic equation: $x^{\log _{3}(x-1)}+2 \cdot(x-1)^{\log _{3} x}=3 \cdot x^{2}$

Selected by Prof. Bara Lajos from "The Most Beautiful Math Problems" by Dan and Vlad Sachelarie.

Grading Rubric

Initial conditions are set: $x>0, x>1, x \neq 1, x-1 \neq 1 \Rightarrow x>1, x \neq 2$ $\qquad$ 1 point

Using the identity: $a^{\log _{c} b}=b^{\log _{c} a}$

the given equation becomes: $x^{\log _{3}(x-1)}+2 \cdot x^{\log _{3}(x-1)}=3 \cdot x^{2}$ $\qquad$ 3 points

$$
x^{\log _{3}(x-1)}=x^{2}
$$

1 point

From the above equation, it follows that $\log _{3}(x-1)=2$ 1 point

With the solution

$$
x_{1}=10
$$

1 point