Write the following as a sum of logarithms:
$$\ln \left(\frac{e^3{x}^4}{y^5}\right)=◻+◻\ln (x)+◻\ln (y)$$

Step 1 ：Write the following as a sum of logarithms: $\ln \left(\frac{e^3{x}^4}{y^5}\right)=◻+◻\ln (x)+◻\ln (y)$

Step 2 ：The question is asking to express the given logarithmic expression as a sum of logarithms. The properties of logarithms can be used to simplify the expression. The properties of logarithms that will be used are:

Step 3 ：1. The logarithm of a quotient is the difference of the logarithms: $\ln (a/b)=\ln (a)-\ln (b)$

Step 4 ：2. The logarithm of a product is the sum of the logarithms: $\ln (ab)=\ln (a)+\ln (b)$

Step 5 ：3. The logarithm of a power is the product of the logarithm and the exponent: $\ln (a^n)=n\ln (a)$

Step 6 ：Using these properties, the given expression can be simplified as follows: $\ln \left(\frac{e^3{x}^4}{y^5}\right)=\ln (e^3)+\ln (x^4)-\ln (y^5)$

Step 7 ：Then, applying the third property of logarithms: $\ln (e^3)+\ln (x^4)-\ln (y^5)=3\ln (e)+4\ln (x)-5\ln (y)$

Step 8 ：Since the natural logarithm of e is 1, the final expression is: $3\ln (e)+4\ln (x)-5\ln (y)=3+4\ln (x)-5\ln (y)$

Step 9 ：Final Answer: $\ln \left(\frac{e^3{x}^4}{y^5}\right)=\boxed{{\displaystyle 3+4\ln (x)-5\ln (y)}}$