express as a single log:

1/2 log x + 3 log y - 4 log x =

log[(y^3)/(x^(7/2))]

To express the given expression as a single logarithm, we can use logarithmic properties.

Let's start by writing the given expression:

1/2 log x + 3 log y - 4 log x

Now, we can use the following logarithmic properties:

1. Product Rule: log(a) + log(b) = log(a × b)
2. Quotient Rule: log(a) - log(b) = log(a / b)
3. Power Rule: log(a^n) = n log(a)

Using these properties, we can simplify the expression:

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

Now, we can use the addition/subtraction property of logarithms:

log(sqrt(x)) + log(y^3) - log(x^4)
= log(sqrt(x) × y^3) - log(x^4)

Finally, we can apply the product rule to combine the logarithms:

log(sqrt(x) × y^3) - log(x^4)
= log(sqrt(x) × y^3 / x^4)

So, the expression 1/2 log x + 3 log y - 4 log x can be written as a single logarithm:

log(sqrt(x) × y^3 / x^4)