The expression in the image is:

$\log_5(x^5) - \log_5(x + 3) + \log_2(x) = N$

To simplify the left-hand side of this equation, we'll apply logarithmic properties:

1. Power Rule: $\log_b(a^c) = c \cdot \log_b(a)$

   • $\log_5(x^5) = 5 \cdot \log_5(x)$

2. Logarithmic Difference Rule: $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$

   • $\log_5(x^5) - \log_5(x+3) = \log_5\left(\frac{x^5}{x+3}\right)$

So, the expression becomes:

$\log_5\left(\frac{x^5}{x+3}\right) + \log_2(x) = N$

This is as simplified as it gets without specific values for $x$ and $N$.

Would you like further details or assistance with this equation?

Here are five related questions you might want to explore:

1. How can I solve for $x$ if $N$ is known?
2. What happens if the bases of the logarithms are the same?
3. How does the change of base formula work for logarithms?
4. Can you explain other logarithmic properties like the product rule?
5. What is the behavior of logarithmic functions as $x$ approaches 0?

Tip: Always remember that the domain of a logarithmic function requires that the argument must be positive.