Solve the following exponential equation and evaluate the expression using logarithms:

Since $4 = 2^2$ and $512 = 2^9$, substitute these into the equation:
$$
(2^2)^{x+1} \times 2^{2x-3} = 2^9
$$

$$
2^{2(x+1)} \times 2^{2x-3} = 2^9
$$
$$
2^{2x+2} \times 2^{2x-3} = 2^9
$$

$$
2^{(2x+2) + (2x-3)} = 2^9
$$
$$
2^{4x-1} = 2^9
$$

$$
4x-1 = 9
$$

$$
4x = 9 + 1
$$
$$
4x = 10
$$
$$
x = \frac{10}{4}
$$
$$
x = \frac{5}{2}
$$
The value of $x$ is $\boxed{\text{2.5}}$.

$$
\log Y = \log \left( \frac{2.735 \times 0.0846}{0.693} \right)
$$

$$
\log Y = \log(2.735) + \log(0.0846) - \log(0.693)
$$

$$
\log(2.735) \approx 0.4376
$$
$$
\log(0.0846) \approx -1.0726 \quad (\text{or } \bar{2}.9274)
$$
$$
\log(0.693) \approx -0.1593 \quad (\text{or } \bar{1}.8407)
$$

$$
\log Y = 0.4376 + (-1.0726) - (-0.1593)
$$
$$
\log Y = 0.4376 - 1.0726 + 0.1593
$$
$$
\log Y = -0.6350 + 0.1593
$$
$$
\log Y = -0.4757
$$

$$
Y = 10^{-0.4757}
$$
$$
Y \approx 0.3344
$$
The evaluated value is $\boxed{\text{0.3344}}$.