If $y$ is a negative number such that $2^{y^2log_35 }$= $5^{log_23}$, then $y$ equals

Question

If the domain of \[ f(x)=\log_{(10x^2-17x+7)}\,(18x^2-11x+1) \] is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then find $90(a+b+c+d+e)$.

Answer 1

Given:

$ 2^{y^2 \log_3 5} = \log_2 3 $

$ \log_2 \left(2^{y^2 \log_3 5}\right) = \log_2 (\log_2 3) $

$ y^2 \log_3 5 = \log_2 (\log_2 3) $

$ \log_3 5 = \frac{\log_2 5}{\log_2 3} $

Substitution yields:

$ y^2 \cdot \frac{\log_2 5}{\log_2 3} = \log_2 (\log_2 3) $

$ y^2 = \frac{\log_2 (\log_2 3) \cdot \log_2 3}{\log_2 5} $

Initial equation: $ 2^{y^2 \log_3 5} = \log_2 3 $

Express the Left Hand Side (LHS) with base 5:

$ 2^{y^2 \log_3 5} = \left(5^{\log_3 2}\right)^{y^2} = 5^{y^2 \log_3 2} $

Equating both sides:

$ 5^{y^2 \log_3 2} = \log_2 3 $

Apply log base 5 to both sides:

$ y^2 \log_3 2 = \log_5 (\log_2 3) $

$ y^2 = \frac{\log_5 (\log_2 3)}{\log_3 2} $

Taking the square root:

$ y = \sqrt{ \frac{\log_5 (\log_2 3)}{\log_3 2} } $

Based on the initial steps, it was determined that:

$ y = \log_2 \left( \frac{1}{3} \right) $

Correct Option: (A) $ \log_2 \left( \frac{1}{3} \right) $

If \(y\) is a negative number such that \(2^{y^2log_35 }\)= \(5^{log_23}\), then \(y\) equals