Let u=ln and v=ln y. Write the expression ln(5√(x3√y)) in terms of u and v. for example, lnx^3y = lnx^3+lny = 3lnx+lny = 3u+v
To write the expression ln(5√(x3√y)) in terms of u and v, we can use the properties of logarithms:
1. Start with the given expression: ln(5√(x3√y)).
2. Simplify the expression inside the logarithm using the properties of radicals. For example, √(x^3) can be written as x^(3/2), and √y can be written as y^(1/2):
ln(5√(x3√y)) = ln(5 * x^(3/2) * y^(1/2)).
3. Apply the logarithmic property that ln(a * b) = ln(a) + ln(b):
ln(5 * x^(3/2) * y^(1/2)) = ln(5) + ln(x^(3/2)) + ln(y^(1/2)).
4. Use the property ln(a * b) = ln(a) + ln(b) to simplify further:
ln(5) + ln(x^(3/2)) + ln(y^(1/2)) = ln(5) + (3/2) * ln(x) + (1/2) * ln(y).
5. Substitute u = ln and v = ln y into the expression:
ln(5) + (3/2) * ln(x) + (1/2) * ln(y) = ln(5) + (3/2) * u + (1/2) * v.
Therefore, the expression ln(5√(x3√y)) in terms of u and v is: ln(5) + (3/2) * u + (1/2) * v.
To write the expression ln(5√(x3√y)) in terms of u and v, we need to find the expression in terms of ln(u) and ln(v).
First, let's simplify the expression step by step:
ln(5√(x3√y))
Using the properties of logarithms, we can rewrite the expression as:
ln(5) + ln(√(x3√y))
Now, let's simplify the second part of the expression:
ln(√(x3√y))
Using the property of logarithms, we can rewrite the expression as:
(1/2)ln(x3√y)
Since u = ln(x) and v = ln(y), we can substitute u and v into the expression:
(1/2)ln(e^3u√e^v)
Using the property of logarithms, we simplify the expression further:
(1/2)*[3u + (1/2)v]
Combining like terms:
(3/2)u + (1/4)v
Therefore, the expression ln(5√(x3√y)) in terms of u and v is (3/2)u + (1/4)v.