To simplify the given expression, let's work step by step using logarithmic properties.

The expression is:

$3\log(x^2y^3) + 4\log\left(\frac{1}{x}\right) - 7\log(y)$

Step 1: Apply logarithmic rules

Expand each term:

1. $3\log(x^2y^3)$:

   • Use the product rule: $\log(AB) = \log(A) + \log(B)$
   • Use the power rule: $a\log(A^b) = ab\log(A)$

   $3\log(x^2y^3) = 3[\log(x^2) + \log(y^3)] = 6\log(x) + 9\log(y)$

2. $4\log\left(\frac{1}{x}\right)$:

   • Use the quotient rule: $\log\left(\frac{A}{B}\right) = \log(A) - \log(B)$
   • $\log(1) = 0$ and simplify:

   $4\log\left(\frac{1}{x}\right) = 4[\log(1) - \log(x)] = -4\log(x)$

3. $-7\log(y)$:

   • This term remains as it is.

   $-7\log(y)$

Step 2: Combine all terms

Now sum up all the simplified terms:

$6\log(x) + 9\log(y) - 4\log(x) - 7\log(y)$

Combine like terms ($\log(x)$ and $\log(y)$):

$(6 - 4)\log(x) + (9 - 7)\log(y) = 2\log(x) + 2\log(y)$

Step 3: Condense further

Use the product rule in reverse ($a\log(A) + b\log(B) = \log(A^aB^b)$):

$2\log(x) + 2\log(y) = \log(x^2) + \log(y^2) = \log(x^2y^2)$

Final Answer:

$\log(x^2y^2)$

Would you like further clarification or details?

Related Questions:

1. What are the key logarithmic properties used in simplifying expressions?
2. How can we expand expressions with logarithms involving roots or powers?
3. How does the logarithm base affect the outcome of simplifications?
4. Can logarithmic expressions be simplified if the base is unknown?
5. What are the practical applications of condensed logarithmic expressions in real-world problems?

Tip:

Always apply logarithmic properties step by step, especially for complex expressions.