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)