This problem involves a sequence of logarithms and the properties of logarithms. In this sequence, each term is the logarithm of a power of \(a\). The progression is \( \log a, \log a^2, \log a^3, \ldots \), which is an arithmetic progression (AP) with a common difference of \( \log a \).

Recall that when you multiply numbers with the same base, you add the exponents. This is the main property of logarithms that we'll use for this problem:

\[ \log_b(mn) = \log_b(m) + \log_b(n) \]

So, to find the sum of the first twenty terms of this sequence, we add them up:

\[ \log a + \log a^2 + \log a^3 + \ldots + \log a^{20} \]

Using the property of logarithms stated above, we can rewrite this sum as:

\[ \log a^{1+2+3+\ldots+20} \]

The sum of the first twenty natural numbers is given by the formula:

\[ S = \frac{n(n+1)}{2} \]

Where \( n \) is the number of terms. Substituting 20 for \( n \), we get:

\[ S = \frac{20(20+1)}{2} = \frac{20 \times 21}{2} = 210 \]

So, the sum of the first twenty terms of the sequence is [ \log a^{210} \]