1/2log(z+1)-2logx-4logy-3logz

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To simplify the given expression, we can use logarithmic rules to combine the terms:

1/2log(z+1) - 2log(x) - 4log(y) - 3log(z)

Using the laws of logarithms:

1/2log(z+1) - log(x^2) - log(y^4) - log(z^3)

Now, applying the power rule of logarithms:

log(sqrt(z+1)) - log(x^2) - log(y^4) - log(z^3)

Combining the logarithmic terms:

log[(sqrt(z+1) / (x^2 y^4 z^3))]

Therefore, the simplified expression for 1/2log(z+1) - 2logx - 4logy - 3logz is:

log[(sqrt(z+1) / (x^2 y^4 z^3))]