If $a^3+b^3=c^3$, then ${\log }_a(c-b)+{\log }_a(c^2+cb+b^2)$ is

We are given the equation: 

a³ + b³ = c³

We need to evaluate the expression:

logₐ(c - b) + logₐ(c² + cb + b²)

Using the property of logarithms, logₐ(x) + logₐ(y) = logₐ(xy), the given expression becomes:

logₐ((c - b)(c² + cb + b²))

Now, observe that:

(c - b)(c² + cb + b²) = c³ - b³

From the given equation, a³ + b³ = c³, we can rewrite it as:

c³ - b³ = a³

Substituting this into the logarithmic expression, we get:

logₐ(c³ - b³) = logₐ(a³)

Using the logarithmic property logₐ(aⁿ) = n, we find:

logₐ(a³) = 3

Therefore, the final result is: 3

The correct option is D.

This solution involves understanding the relationship between a³, b³, and c³, applying logarithmic properties effectively, and substituting the given equation to arrive at the result.