Write the following as a sum and/or difference of logarithms

log((∛x^2+1)/(x^2-1))

1/3 log(x^2+1) - log(x^2-1)

To express log((∛x^2+1)/(x^2-1)) as a sum or difference of logarithms, we can use the properties of logarithms.

Using the quotient rule for logarithms, we can write log((∛x^2+1)/(x^2-1)) as the difference of logarithms:

log((∛x^2+1)/(x^2-1)) = log(∛x^2+1) - log(x^2-1)

Now, let's simplify each part:

1. log(∛x^2+1):
To simplify this logarithm, we can use the property that log(a^n) = n log(a). Here, we have a cube root, so we can rewrite it as (∛x^2+1)^(1/3):

log(∛x^2+1) = log((∛x^2+1)^(1/3)) = (1/3) log(∛x^2+1)

2. log(x^2-1):
No simplification is possible for this logarithm as it stands.

Therefore, the final representation of log((∛x^2+1)/(x^2-1)) as a sum and/or difference of logarithms is:

log((∛x^2+1)/(x^2-1)) = (1/3) log(∛x^2+1) - log(x^2-1)