We are asked to find the value of $w$ given the equation $\log(w+5) + \log(w-5) = 4\log(2) + 2\log(3)$.

1. Problem Description

We are asked to find the value of $w$ given the equation $\log(w+5) + \log(w-5) = 4\log(2) + 2\log(3)$.

2. Solution Steps

We can simplify the given equation using logarithm properties.

First, we use the property $\log(a) + \log(b) = \log(ab)$ on the left-hand side:

$\log((w+5)(w-5)) = 4\log(2) + 2\log(3)$

$\log(w^2 - 25) = 4\log(2) + 2\log(3)$

Next, we use the property $n\log(a) = \log(a^n)$ on the right-hand side:

$\log(w^2 - 25) = \log(2^4) + \log(3^2)$

$\log(w^2 - 25) = \log(16) + \log(9)$

Using the property $\log(a) + \log(b) = \log(ab)$ again:

$\log(w^2 - 25) = \log(16 \times 9)$

$\log(w^2 - 25) = \log(144)$

Since the logarithms are equal, the arguments must be equal:

$w^2 - 25 = 144$

$w^2 = 144 + 25$

$w^2 = 169$

$w = \pm\sqrt{169}$

$w = \pm 13$

However, we must check if the solutions are valid. Since we have $\log(w+5)$ and $\log(w-5)$, we require $w+5 > 0$ and $w-5 > 0$.

This implies $w > -5$ and $w > 5$. Therefore, we must have $w > 5$.

If $w = 13$, then $w+5 = 18 > 0$ and $w-5 = 8 > 0$, so $w=13$ is a valid solution.

If $w = -13$, then $w+5 = -8 < 0$ and $w-5 = -18 < 0$, so $w=-13$ is not a valid solution.

Therefore, the only solution is $w = 13$.

3. Final Answer

$w = 13$