2logt + 1/4[logt + 3logp]

To simplify the expression 2logt + 1/4[logt + 3logp], we'll follow these steps:

Step 1: Apply the product rule of logarithms.
The product rule states that loga(b) + loga(c) = loga(bc). So, let's rewrite the expression using this rule:
2logt + 1/4[logt + 3logp] = logt^2 + logt^(1/4) + logp^(3/4)

Step 2: Combine the logarithms using the power rule.
The power rule states that loga(b^p) = p * loga(b). Let's apply this rule to combine the logarithms:
logt^2 + logt^(1/4) + logp^(3/4) = log(t^2 * t^(1/4) * p^(3/4))

Step 3: Simplify the expression further, if possible.
The expression log(t^2 * t^(1/4) * p^(3/4)) can still be simplified by applying the properties of exponents. Let's do this:
t^2 * t^(1/4) * p^(3/4) = t^(2 + 1/4) * p^(3/4) = t^(9/4) * p^(3/4)

Step 4: Rewrite the expression in a simplified form.
Now that we have simplified the expression, we can rewrite it as follows:
2logt + 1/4[logt + 3logp] = log(t^(9/4) * p^(3/4))

So, the simplified form of the expression 2logt + 1/4[logt + 3logp] is log(t^(9/4) * p^(3/4)).