To express the given expression as a single logarithm, we'll follow these steps:

1. Apply the logarithm property:

   • Use the property $a \log_b(c) = \log_b(c^a)$ to move the coefficients inside the logarithms as exponents.

   $5 \log(2) = \log(2^5)$ $3 \log(3) = \log(3^3)$

2. Rewrite the expression: $\log(2^5) + \log(3^3) - 2$

   Since $\log(2^5) + \log(3^3)$ is a sum of logarithms, we can use the property $\log(a) + \log(b) = \log(a \cdot b)$ to combine them:

   $\log(2^5 \cdot 3^3) - 2$

3. Simplify: $2^5 = 32 \quad \text{and} \quad 3^3 = 27$

   So the expression becomes: $\log(32 \cdot 27) - 2 = \log(864) - 2$

4. Final step: We still have $-2$ outside the logarithm. To combine this, express $-2$ as $\log(10^2)$, because $\log(10^2) = 2$. Thus, we have:

   $\log(864) - \log(10^2)$

   Using the property $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$, this simplifies to: $\log\left(\frac{864}{10^2}\right) = \log\left(\frac{864}{100}\right)$

   Finally: $\log(8.64)$

So, the expression simplifies to $\log(8.64)$.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do logarithmic properties help in simplifying expressions?
2. What is the inverse of logarithmic operations?
3. Can you express $7\log(4) + \log(5) - 4$ as a single logarithm?
4. How would you solve an equation involving $\log(x) + \log(2x) = 3$?
5. What are the different bases of logarithms and how do they affect the result?

Tip: When simplifying logarithmic expressions, remember that addition inside the logarithms corresponds to multiplication of arguments, and subtraction corresponds to division.