Condense the expression: 1/5log3(32) - 2log3x + 1/2log3y

I can't put the 3 as a small number on the bottom on all 3 parts of the trinomial.

To condense the expression 1/5log₃(32) - 2log₃x + 1/2log₃y, you can use the properties of logarithms to combine the terms with the same base, which is 3 in this case.

Let's start by simplifying each term individually:

1/5log₃(32) can be rewritten as log₃(32^(1/5)), using the property logₐ(b) = logₐ(c^d) = d * logₐ(c). Simplifying further, 32^(1/5) is equal to the 5th root of 32, which is 2. So, 1/5log₃(32) becomes log₃(2).

Next, we have -2log₃x, which can be written as log₃(x^-2) using the property logₐ(b) = -logₐ(c^-d) = -d * logₐ(c). Simplifying further, x^-2 is equal to 1/x^2. So, -2log₃x becomes -2log₃(1/x^2), which can be further rewritten as -log₃((1/x^2)^2), using the property logₐ(b) = logₐ(c^d) = d * logₐ(c). Finally, this simplifies to -log₃(1/x^4).

Lastly, we have 1/2log₃y, which can be written as log₃(y^(1/2)), using the property logₐ(b) = logₐ(c^d) = d * logₐ(c). Simplifying further, y^(1/2) is equal to the square root of y. So, 1/2log₃y becomes log₃(√y).

Now that we have simplified each term individually, we can combine them:

log₃(2) - log₃(1/x^4) + log₃(√y).

To condense this further, we can apply the properties of logarithms related to multiplication and division. Specifically, if we have logₐ(b) - logₐ(c), it is equivalent to logₐ(b/c).

Applying this property, we can rewrite the expression as a single logarithm:

log₃(2 * √y / (1/x^4))

Simplifying further:

log₃(2 * √y * x^4).

Therefore, the condensed form of the expression is log₃(2√y * x^4).