The value of `log_(10)1 + log_(10) 10 + log_(10)100+`…….. `log_(10)(10000000000)`

Options

A. 10
B. 11
C. 11111111111
D. 55

Answer

To solve the problem, we need to evaluate the expression:

\[ \log_{10}1 + \log_{10}10 + \log_{10}100 + \log_{10}1000 + \ldots + \log_{10}(10000000000) \]

### Step 1: Identify the terms in the series
The series consists of logarithms of powers of 10. We can express each term as follows:
- \(\log_{10}1 = \log_{10}(10^0) = 0\)
- \(\log_{10}10 = \log_{10}(10^1) = 1\)
- \(\log_{10}100 = \log_{10}(10^2) = 2\)
- \(\log_{10}1000 = \log_{10}(10^3) = 3\)
- ...
- \(\log_{10}(10000000000) = \log_{10}(10^{10}) = 10\)

### Step 2: Write the series in terms of powers
The series can be rewritten as:
\[ 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 \]

### Step 3: Use the formula for the sum of the first n natural numbers
The sum of the first \(n\) natural numbers is given by the formula:
\[ S_n = \frac{n(n + 1)}{2} \]
In our case, \(n = 10\).

### Step 4: Calculate the sum
Substituting \(n = 10\) into the formula:
\[ S_{10} = \frac{10(10 + 1)}{2} = \frac{10 \times 11}{2} = \frac{110}{2} = 55 \]

### Conclusion
Thus, the value of the expression \(\log_{10}1 + \log_{10}10 + \log_{10}100 + \ldots + \log_{10}(10000000000)\) is:
\[ \boxed{55} \]

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