Simplify the equation using laws of logarithms. 1/2log4 (x+3) - log4 (x) + log4 (3)

Sure, let's simplify the equation step by step using the laws of logarithms.

Given equation: 1/2log4 (x+3) - log4 (x) + log4 (3)

Step 1: Apply the power rule of logarithms, which states that logb (a^n) = n * logb (a).

So, 1/2log4 (x+3) becomes log4 √(x+3).

Now, the equation becomes: log4 √(x+3) - log4 (x) + log4 (3)

Step 2: Apply the quotient rule of logarithms, which states that logb (a) - logb (c) = logb (a/c).

So, log4 √(x+3) - log4 (x) becomes log4 [√(x+3) / x].

Now, the equation becomes: log4 [√(x+3) / x] + log4 (3)

Step 3: Apply the product rule of logarithms, which states that logb (a) + logb (c) = logb (a*c).

So, log4 [√(x+3) / x] + log4 (3) becomes log4 [3 * √(x+3) / x].

So, the simplified form of the given equation is: log4 [3 * √(x+3) / x]