First we must understand how to write a logarithmic function:
![log_{b}a=x](https://tex.z-dn.net/?f=log_%7Bb%7Da%3Dx)
In the equation above, b is the base, x is the exponent, and a is the answer. These same variables can be rearranged to be expressed as an exponential equation as followed:
![b^x=a](https://tex.z-dn.net/?f=b%5Ex%3Da)
Next, we need to understand basic logarithm rules.
1. When a value is raised to a power, we can move the exponent to the front of the logarithm. Example:
log(a^2) = 2log(a)
2. When two variables are multiplied together, we can add the logarithms of the individual variables together. Example:
log(ab) = log(a) + log(b)
3. When a variable is divided by another variable, we can subtract the logarithms of the individual variables. Example:
log(a/b) = log(a) - log(b)
Now we can use these rules to solve the problem.
![log(r)=log( \sqrt[3]{ \frac{A^2B}{C} } )](https://tex.z-dn.net/?f=log%28r%29%3Dlog%28%20%5Csqrt%5B3%5D%7B%20%5Cfrac%7BA%5E2B%7D%7BC%7D%20%7D%20%29)
We can rewrite the cube root as:
![log(r) = log( (\frac{A^2B}{C})^ \frac{1}{3} )](https://tex.z-dn.net/?f=log%28r%29%20%3D%20log%28%20%28%5Cfrac%7BA%5E2B%7D%7BC%7D%29%5E%20%5Cfrac%7B1%7D%7B3%7D%20%29%20)
Now we can move the one-third to the front:
![log(r) = \frac{1}{3} log( \frac{A^2B}{C} )](https://tex.z-dn.net/?f=log%28r%29%20%3D%20%20%5Cfrac%7B1%7D%7B3%7D%20log%28%20%5Cfrac%7BA%5E2B%7D%7BC%7D%20%29)
Now we can split up the logarithm:
![log(r) = \frac{1}{3} (log(A^2)+log(B)-log(C))](https://tex.z-dn.net/?f=log%28r%29%20%3D%20%20%5Cfrac%7B1%7D%7B3%7D%20%28log%28A%5E2%29%2Blog%28B%29-log%28C%29%29)
Finally, we can move the exponent to the front of the log of A:
![log(r) = \frac{1}{3} (2log(A)+log(B)-log(C))](https://tex.z-dn.net/?f=log%28r%29%20%3D%20%5Cfrac%7B1%7D%7B3%7D%20%282log%28A%29%2Blog%28B%29-log%28C%29%29)
Distribute the one-third to get the answer:
The answer is (4).