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 properties of logarithms to simplify it.

We'll start by using the power rule of logarithms, which states that log_a(x^m) = m*log_a(x). Applying this rule to each term in the equation, we get:

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

Next, we can use the product rule of logarithms, which states that log_a(x * y) = log_a(x) + log_a(y). Applying this rule to combine the terms, we get:

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

Now, we can simplify the expression inside the logarithm:

log_3(x) = log_3((49 * 27^(2/3)) / (16^(3/2)))

Using the power rule again, we can rewrite the expression inside the logarithm:

log_3(x) = log_3((49 * (3^3)^(2/3)) / (2^4)^(3/2))

Simplifying further:

log_3(x) = log_3((49 * 3^(2/3) * 3^(2/3)) / (2^3 * 2^3))

Using the properties of exponents, we can simplify 3^(2/3) as (3^(1/3))^2 = (cubic root of 3)^2:

log_3(x) = log_3((49 * (cubic root of 3)^2 * (cubic root of 3)^2) / (2^3 * 2^3))

Simplifying the expression inside the logarithm again:

log_3(x) = log_3((49 * 3 * 3) / (2^3 * 2^3))

log_3(x) = log_3((49 * 9) / (8 * 8))

Simplifying further:

log_3(x) = log_3(441/64)

Now, we can equate the exponent of the logarithm with the value of x:

x = 441/64

Therefore, the solution to the equation log_3(x) = 2 * log_3(7) + 2/3 * log_3(27) - 3/2 * log_3(16) is x = 441/64.