`log_(10)(log_(2)3) + log_(10)(log_(3)4) + …………+log_(10)(log_(1023)1024)` simplies to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the expression \( \log_{10}(\log_{2}3) + \log_{10}(\log_{3}4) + \ldots + \log_{10}(\log_{1023}1024) \), we can follow these steps:

### Step 1: Rewrite the expression using properties of logarithms
We know that the sum of logarithms can be expressed as the logarithm of the product. Therefore, we can rewrite the expression as:

\[ \log_{10}(\log_{2}3 \cdot \log_{3}4 \cdot \ldots \cdot \log_{1023}1024) \]

### Step 2: Simplify each logarithm
Using the change of base formula, we can express each logarithm in the product as:

\[ \log_{b}a = \frac{\log_{10}a}{\log_{10}b} \]

Thus, we can rewrite the product:

\[ \log_{2}3 = \frac{\log_{10}3}{\log_{10}2}, \quad \log_{3}4 = \frac{\log_{10}4}{\log_{10}3}, \quad \ldots, \quad \log_{1023}1024 = \frac{\log_{10}1024}{\log_{10}1023} \]

### Step 3: Combine the products
Putting it all together, we have:

\[ \log_{2}3 \cdot \log_{3}4 \cdots \log_{1023}1024 = \frac{\log_{10}3}{\log_{10}2} \cdot \frac{\log_{10}4}{\log_{10}3} \cdots \frac{\log_{10}1024}{\log_{10}1023} \]

Notice that in this product, all terms will cancel except for the first denominator and the last numerator:

\[ = \frac{\log_{10}1024}{\log_{10}2} \]

### Step 4: Substitute back into the logarithm
Now we substitute this back into our logarithm:

\[ \log_{10}\left(\frac{\log_{10}1024}{\log_{10}2}\right) \]

### Step 5: Simplify further
We know that \( 1024 = 2^{10} \), thus:

\[ \log_{10}1024 = \log_{10}(2^{10}) = 10 \cdot \log_{10}2 \]

Substituting this into our expression gives:

\[ \log_{10}\left(\frac{10 \cdot \log_{10}2}{\log_{10}2}\right) = \log_{10}(10) \]

### Step 6: Final simplification
Finally, we know that:

\[ \log_{10}(10) = 1 \]

Thus, the entire expression simplifies to:

\[ \boxed{1} \]