Assume that x, y, z and b are positive numbers. Use the properties of logarithms to write the expression logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z.

logb(4) +logb((x^7 y^2)/(z^4))^.5

logb(4) +logb((x^7 y^2)/(z^4))^.5

logb(4)+3.5 logb(x) + logb (y) - 2 logb (z)

To write the expression logb(4) sqrt((x^7 y^2)/(z^4)) using the properties of logarithms and in terms of the logarithms x, y, and z, we can start by applying the power rule and the square root property of logarithms.

1. Power Rule: logb(a^m) = m * logb(a)
This rule allows us to bring the exponent of a logarithm down as a coefficient.

Using the power rule, we can rewrite the expression as:

logb(4) + 1/2 * logb((x^7 y^2)/(z^4))

2. Square Root Property: logb(sqrt(a)) = logb(a^(1/2)) = 1/2 * logb(a)
This property allows us to express the square root of a value as a power that we can apply the power rule to.

Applying the square root property, we rewrite the expression further:

logb(4) + 1/2 * [logb(x^7 y^2) - logb(z^4)]

Now, let's simplify the logarithms using the properties of logarithms:

Using the product rule: logb(a * b) = logb(a) + logb(b)

We can rewrite logb(x^7 y^2) as:
logb(x^7) + logb(y^2)

Using the power rule: logb(a^m) = m * logb(a)

We can further simplify as:
7 * logb(x) + 2 * logb(y)

Lastly, applying the power rule to logb(z^4):
4 * logb(z)

Combining all the simplified expressions, we get:

logb(4) + 1/2 * [(7 * logb(x) + 2 * logb(y)) - 4 * logb(z)]

Therefore, the expression logb(4) sqrt((x^7 y^2)/(z^4)) in terms of the logarithms x, y, and z is:

logb(4) + 1/2 * [(7 * logb(x) + 2 * logb(y)) - 4 * logb(z)]