Let's solve the given equation:

$y = 2\log_3(5^c) - \log_3(4^x)$

Step 1: Use logarithmic properties

We will apply the following logarithmic properties:

• $\log_b(a^n) = n\log_b(a)$
• $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$

Step 2: Simplify using properties

1. $\log_3(5^c) = c\log_3(5)$, so $2\log_3(5^c) = 2c\log_3(5)$.
2. $\log_3(4^x) = x\log_3(4)$.

Substituting these into the equation:

$y = 2c\log_3(5) - x\log_3(4)$

Step 3: Final equation

The equation simplifies to:

$y = 2c\log_3(5) - x\log_3(4)$

This is a linear equation in terms of $c$ and $x$, where the coefficients are $\log_3(5)$ and $\log_3(4)$.

Let me know if you'd like any further explanation or if you have specific questions! Here's some related practice:

1. How do we solve logarithmic equations like this for $x$ or $c$?
2. What would happen if we set $y = 0$ in this equation?
3. How can logarithmic identities be used to solve more complex expressions?
4. Can we convert logarithms to exponentials to simplify the equation further?
5. How would this equation change if we were dealing with logarithms in a different base?

Tip: When simplifying logarithmic expressions, always look for opportunities to apply properties like $\log_b(a^n) = n\log_b(a)$ and $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$ to reduce the complexity.