The given expression is:

$4(\log_c x - 5\log_c z) + 3\log_c y$

We will simplify it step by step into a single logarithmic expression.

Step 1: Apply the logarithmic properties inside the parentheses

First, use the property that $a \log_b M = \log_b M^a$ to move the coefficients inside the logs as exponents:

$4 \left( \log_c x - \log_c z^5 \right) + 3 \log_c y$

Step 2: Apply the subtraction rule

The subtraction inside the logarithm, $\log_b A - \log_b B = \log_b \left( \frac{A}{B} \right)$, can now be applied:

$4 \log_c \left( \frac{x}{z^5} \right) + 3 \log_c y$

Step 3: Apply the coefficient as an exponent

Now, apply the coefficient 4 to the logarithm, using the same exponent property again:

$\log_c \left( \frac{x}{z^5} \right)^4 + 3 \log_c y$

This simplifies to:

$\log_c \left( \frac{x^4}{z^{20}} \right) + 3 \log_c y$

Step 4: Apply the addition rule

Finally, use the addition rule $\log_b A + \log_b B = \log_b (A \cdot B)$ to combine the terms:

$\log_c \left( \frac{x^4}{z^{20}} \cdot y^3 \right)$

Final Answer:

The expression written as a single logarithm is:

$\log_c \left( \frac{x^4 y^3}{z^{20}} \right)$

Let me know if you need more details on any step!

Here are 5 related questions:

1. How do you combine multiple logarithmic terms with both addition and subtraction?
2. Can you explain the power rule in logarithms more deeply?
3. What is the inverse operation of logarithms?
4. How does changing the base of a logarithm affect its properties?
5. How do you simplify a logarithmic equation involving multiple variables?

Tip: Always remember the key logarithmic properties: the product, quotient, and power rules, as they help in combining or simplifying logarithmic expressions.