Log_3(x) = 2 * log_3(7) + 2/3 * log_3(27) - 3/2 * log_3(16)

To solve the equation log_3(x) = 2 * log_3(7) + 2/3 * log_3(27) - 3/2 * log_3(16), we can use the logarithmic properties:

1. log_b(a) + log_b(c) = log_b(a * c)
2. log_b(a) - log_b(c) = log_b(a / c)
3. log_b(a^c) = c * log_b(a)

Using these properties, let's simplify the equation step by step:

log_3(x) = 2 * log_3(7) + 2/3 * log_3(27) - 3/2 * log_3(16)

Applying the first property to the right-hand side:

log_3(x) = log_3(7^2) + log_3(27^(2/3)) - log_3(16^(3/2))

log_3(x) = log_3(49) + log_3(9) - log_3(64)

Applying the third property to each term:

log_3(x) = log_3(49 * 9 / 64)

log_3(x) = log_3(441 / 64)

Now, since the equation is in logarithmic form, we can convert it into exponential form:

x = 441 / 64

Therefore, the solution to the equation is x = 441 / 64.