Let u=lnx and v=lny. Write the expression ln3√x/2y in terms of u and v. For example, lnx^3y=lnx^3+lny=3lnx +lny= 3u+v.
To write the expression ln(3√(x/2y)) in terms of u and v, we need to simplify it using the properties of logarithms.
First, let's express √(x/2y) in terms of u and v. We can do this by using the property that ln(a/b) = ln(a) - ln(b):
ln(√(x/2y)) = ln(x/2y)^(1/2) = (1/2)ln(x/2y)
Next, let's write ln(3√(x/2y)) using u and v:
ln(3√(x/2y)) = ln(3 * √(x/2y))
= ln(3) + ln(√(x/2y))
= ln(3) + ln(x/2y)^(1/2)
= ln(3) + (1/2)ln(x/2y)
Now, let's express ln(x/2y) using u and v. Using the property that ln(ab) = ln(a) + ln(b), and the fact that u = ln(x) and v = ln(y), we can write:
ln(x/2y) = ln(x) - ln(2y)
= u - ln(2) - ln(y)
= u - v - ln(2)
Substituting this back into the expression, we get:
ln(3√(x/2y)) = ln(3) + (1/2)(u - v - ln(2))
= ln(3) + (1/2)u - (1/2)v - (1/2)ln(2)
So, the expression ln3√(x/2y) in terms of u and v is ln(3) + (1/2)u - (1/2)v - (1/2)ln(2).