The equation given is:

$\log_2(x + 3) = \log_2(x - 3) + \log_4(16) + 5 \log_5(3)$

We need to simplify and solve for $x$.

Step 1: Simplify the logarithmic terms

1. $\log_4(16)$: Recall that $16 = 4^2$, so $\log_4(16) = 2$

2. $5 \log_5(3)$: Using the logarithmic property $a \log_b(c) = \log_b(c^a)$, this simplifies to: $5 \log_5(3) = \log_5(3^5) = \log_5(243)$

Step 2: Substitute these into the equation

Now the equation becomes: $\log_2(x + 3) = \log_2(x - 3) + 2 + \log_5(243)$

Step 3: Eliminate logarithmic terms by equating arguments

Since the equation is expressed in terms of logarithms, we'll start by isolating the logarithmic expressions with base 2.

$\log_2(x + 3) - \log_2(x - 3) = 2$

Using the property of logarithms: $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$, we have: $\log_2\left(\frac{x + 3}{x - 3}\right) = 2$

Step 4: Convert from logarithmic form to exponential form

Using the definition of a logarithm: $\log_b(a) = c \Rightarrow b^c = a$, convert this to exponential form: $\frac{x + 3}{x - 3} = 2^2 = 4$

Step 5: Solve the resulting equation

Now solve: $\frac{x + 3}{x - 3} = 4$ Multiply both sides by $x - 3$: $x + 3 = 4(x - 3)$ Expand the right-hand side: $x + 3 = 4x - 12$ Now, solve for $x$: $3 + 12 = 4x - x$ $15 = 3x$ $x = 5$

Final Answer:

The solution is $x = 5$.

Would you like a detailed explanation of any step or further assistance?

Here are some additional related questions for further practice:

1. How would the solution change if the logarithms had different bases?
2. What are the domain restrictions for this logarithmic equation?
3. Can you solve an equation with logarithms of different bases without converting them to the same base?
4. How do the properties of logarithms simplify solving equations?
5. What is the general approach to solving exponential equations?

Tip: Always check the domain restrictions of logarithmic functions to avoid invalid solutions (e.g., arguments inside logs must be positive).