`log_(10) 10+log_(10) 100 +log_(10) 1000+ log_(10) 10000+ log_(10)100000 ` is equals to

Options

A. `15`
B. `log11111`
C. `log_(10) 1111`
D. `14 log_(10) 100`

Answer

To solve the expression \( \log_{10} 10 + \log_{10} 100 + \log_{10} 1000 + \log_{10} 10000 + \log_{10} 100000 \), we will use the properties of logarithms step by step.

### Step 1: Break down each logarithm
We know that:
- \( \log_{10} 10 = 1 \)
- \( \log_{10} 100 = \log_{10} (10^2) = 2 \cdot \log_{10} 10 = 2 \)
- \( \log_{10} 1000 = \log_{10} (10^3) = 3 \cdot \log_{10} 10 = 3 \)
- \( \log_{10} 10000 = \log_{10} (10^4) = 4 \cdot \log_{10} 10 = 4 \)
- \( \log_{10} 100000 = \log_{10} (10^5) = 5 \cdot \log_{10} 10 = 5 \)

### Step 2: Substitute the values
Now we can substitute the values we found into the original expression:
\[ \log_{10} 10 + \log_{10} 100 + \log_{10} 1000 + \log_{10} 10000 + \log_{10} 100000 = 1 + 2 + 3 + 4 + 5 \]

### Step 3: Add the values
Now, we add these values together:
\[ 1 + 2 + 3 + 4 + 5 = 15 \]

### Conclusion
Thus, the value of the expression \( \log_{10} 10 + \log_{10} 100 + \log_{10} 1000 + \log_{10} 10000 + \log_{10} 100000 \) is \( \boxed{15} \).

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