If y is a negative number such that $2^{y^2log_3 5} = 5^{\log_2 3}$, then y equals

The equation we need to solve is:

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

where $y$ is a negative number.

---

Using the property $a^{\log_b c} = c^{\log_b a}$, we can rewrite the right side:

$5^{\log_2 3} = 3^{\log_2 5}$

Our equation becomes:

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

---

Taking $\log_2$ of both sides:

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

Using the power rule $\log_a(b^c) = c\log_a(b)$:

$y^2\log_3 5 = \log_2 5 \cdot \log_2 3$

---

Using change of base formula:

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

Substituting:

$y^2 \cdot \dfrac{\log_2 5}{\log_2 3} = \log_2 5 \cdot \log_2 3$

---

Solving for $y^2$:

$y^2 = \dfrac{\log_2 5 \cdot \log_2 3 \cdot \log_2 3}{\log_2 5}$

The $\log_2 5$ terms cancel:

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

---

Taking the square root:

$y = \pm \log_2 3$

Since we're told $y$ is negative:

$y = -\log_2 3$