The expression you provided is:

$2\log(x) - 4\log(y)$

We can simplify this expression using the properties of logarithms. Here are the steps:

1. Apply the power rule of logarithms, which states that $a\log(b) = \log(b^a)$.

So:

$2\log(x) = \log(x^2)$ and $4\log(y) = \log(y^4)$

2. Now, substitute these into the original expression:

$\log(x^2) - \log(y^4)$

3. Use the subtraction rule for logarithms: $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$.

Thus, the expression becomes:

$\log\left(\frac{x^2}{y^4}\right)$

So, the simplified form of the given expression is:

$\log\left(\frac{x^2}{y^4}\right)$

If you'd like more details on any of the logarithmic properties or have additional questions, feel free to ask!

Related questions:

1. How do logarithmic properties like the product rule or quotient rule work?
2. How can logarithms be used to solve equations involving exponents?
3. Can you explain the natural logarithm ($\ln$) and its properties?
4. What is the inverse relationship between logarithms and exponentials?
5. How do you solve logarithmic equations like $\log(x) = 2$?

Tip:

When simplifying logarithmic expressions, always remember to look for opportunities to apply the power, product, and quotient rules to condense them into simpler forms.