Question

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

Answer 1

 1/2log(z+1)-2logx-4logy-3logz=  log(√(z+1)/x²у⁴z³)

Answer 2

To simplify the expression, we'll use logarithmic properties. Remember that for any positive base "b," the following properties apply:

1. log_b(xy) = log_b(x) + log_b(y)
2. log_b(x/y) = log_b(x) - log_b(y)
3. log_b(x^n) = n * log_b(x)

Now, let's simplify the expression step by step:

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

Step 1: Use property 3 to bring the coefficients into the logarithms: log((z + 1)^(1/2)) - log(x^2) - log(y^4) - log(z^3)

Step 2: Use property 1 to combine the logarithms: log((z + 1)^(1/2) / (x^2 * y^4 * z^3))

The final simplified expression is: log((z + 1)^(1/2) / (x^2 * y^4 * z^3))