To prove the identity:

$\log_a(xy) = \log_a x + \log_a y,$

we will use properties of logarithms and exponents.

Step-by-step proof:

1. Definition of logarithm: A logarithm is the inverse operation of exponentiation. If $\log_a M = N$, then: $a^N = M.$ In this case, we use the fact that $\log_a x = p$ implies $a^p = x$ and $\log_a y = q$ implies $a^q = y$.

2. Multiplication property: Now, consider the logarithm of the product $xy$: $\log_a(xy).$ Using the property of exponents: $a^{\log_a(xy)} = xy.$

3. Substituting logarithmic values: From the definition of logarithms: $xy = a^p \cdot a^q.$ By the property of exponents, multiplying terms with the same base: $a^p \cdot a^q = a^{p+q}.$

4. Taking logarithms: Since $\log_a(a^z) = z$ for any $z$, applying the logarithm to both sides of $a^{p+q} = xy$, we get: $\log_a(xy) = p + q.$ But from earlier, we know $p = \log_a x$ and $q = \log_a y$, so: $\log_a(xy) = \log_a x + \log_a y.$

Thus, we have proved the logarithmic property.

Would you like more details or have any questions on this proof?

Here are 5 related questions to deepen understanding:

1. What are the other key logarithmic properties, such as for division or powers?
2. How does the change of base formula work for logarithms?
3. Can logarithms of negative numbers be defined in the real number system?
4. How would this proof change if we were dealing with natural logarithms ($\ln$)?
5. What is the relationship between logarithms and exponents in exponential growth models?

Tip: Whenever working with logarithms, always remember that logarithmic identities are derived directly from the properties of exponents.