if [.] denotes the greatest integer function then the value of natural number n satisfying the equation [log 1/log2] + [log2/log2] + [log3/log2]+......+ [log n/log2] = 1538 is.....

Let’s dive into the problem you're facing. We need to find the natural number \( n \) that satisfies the equation: \[\left[\frac{\log 1}{\log 2}\right] + \left[\frac{\log 2}{\log 2}\right] + \left[\frac{\log 3}{\log 2}\right] + \ldots + \left[\frac{\log n}{\log 2}\right] = 1538\]Here, the notation \([x]\) represents the greatest integer function (also known as the floor function), which means we take the largest integer less than or equal to \( x \). Let's break this down step by step.

Understanding the Terms

We know that \(\log_k x\) represents the logarithm of \( x \) with base \( k \). In this case, we are using base 2. Therefore, the expression \(\frac{\log k}{\log 2}\) actually gives us the logarithm of \( k \) in base 2, which can also be written as \(\log_2 k\). This simplifies our equation to:

\[\left[\log_2 1\right] + \left[\log_2 2\right] + \left[\log_2 3\right] + \ldots + \left[\log_2 n\right] = 1538\]

Calculating Each Term

Now, let’s calculate the values of \(\left[\log_2 k\right]\) for various \( k \). The value of \(\left[\log_2 k\right]\) represents how many times we can double 1 before exceeding \( k \). For example:

Pattern Recognition

From this, we observe a pattern. The value of \(\left[\log_2 k\right]\) remains constant within certain ranges of \( k \):- From \( k = 1 \) to \( k = 2 \), the result is \( 0 \).- From \( k = 3 \) to \( k = 4 \), the result is \( 1 \).- From \( k = 5 \) to \( k = 8 \), the result is \( 2 \).- From \( k = 9 \) to \( k = 16 \), the result is \( 3 \), and so forth.In general, for \( k \) in the range \( [2^m, 2^{m+1}-1] \), the value of \(\left[\log_2 k\right]\) is \( m \). Each \( m \) contributes \( m \) to the sum for \( 2^m \) values of \( k \). Therefore, the contribution to the sum from each interval can be calculated as follows:

Summing Contributions

Now, let’s sum these contributions:

This means that up until \( k = 128 \) (which is \( 2^7 \)), we have \( 1537 \) as our total. To satisfy the equation, we need to add one more to reach \( 1538 \). The next number is \( 129 \), which will still fall under the \( 7 \)-range, meaning:

\[\left[\log_2 129\right] = 7\]Thus, we conclude that \( n \) must be \( 129 \) to satisfy the original equation. Therefore, the value of \( n \) is:

Final Answer

The required natural number n is 129.