To expand this logarithm, you must use every rule found in the link below!

We can start by separating the division term by using subtraction:

`log_c (root(3)(ac^2)/(bsqrtd)) = log_c root(3)(ac^2) - log_c bsqrtd`

Now, we want to get all of the root terms into exponential forms in order to use the power rule for logarithms. Recall, `root(n)(x) = x^(1/n)` :

`log_c root(3)(ac^2) - log_c bsqrtd = log_c a^(1/3)c^(2/3) - log_c bd^(1/2)`

Now, we can separate each term first by using the multiplication rule of logarithms:

`log_c a^(1/3)c^(2/3) - log_c bd^(1/2) = log_c a^(1/3) + log_c c^(2/3) - (log_c b + log_c d^(1/2))`

Before we continue, let's distribute that negative sign to avoid mistakes!

`=log_c a^(1/3) + log_c c^(2/3) - log_c b - log_c d^(1/2)`

We can continue by using the power rule of logarithms:

`= 1/3 log_c a + 2/3 log_c c - log_c b - 1/2 log_c d`

Finally, notice that `log_c c` term can be reduced to 1 based on the definition of logarithms:

`= 1/3 log_c a + 2/3 - log_c b - 1/2 log_c d`

If you wanted to enter this into a calculator, you might want to use change of base, because `log_c` generally is not found in your set of buttons! Let's change base to `e`, giving us natural logarithms:

`= 1/3 ln a/(ln c) + 2/3 - lnb/lnc - 1/2 ln d/ln c`

Here, we have the most expanded form possible of the above problem. Notice that we used the division, multiplication, and power rules as well as change-of-base! Also, note that power rules are equivalent to root rules.

I hope this helps!