`log_(10)(log_(2)3) + log_(10)(log_(3)4) + …….. + log_(10) (log_(1023) 1024)` equals

Options

A. 10
B. e
C. 1
D. 0

Answer

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

### Step 1: Rewrite the Sum
We can express the entire sum as:
\[ \log_{10}(\log_{2}3) + \log_{10}(\log_{3}4) + \ldots + \log_{10}(\log_{1023}1024) = \log_{10}(\log_{2}3 \cdot \log_{3}4 \cdots \log_{1023}1024) \]

**Hint:** Use the property of logarithms that states \( \log_a b + \log_a c = \log_a(b \cdot c) \).

### Step 2: Simplify the Product Inside the Logarithm
Now we need to simplify the product:
\[ \log_{2}3 \cdot \log_{3}4 \cdots \log_{1023}1024 \]
Using the change of base formula, we can rewrite each logarithm:
\[ \log_{2}3 = \frac{\log 3}{\log 2}, \quad \log_{3}4 = \frac{\log 4}{\log 3}, \quad \ldots, \quad \log_{1023}1024 = \frac{\log 1024}{\log 1023} \]

**Hint:** Remember that \( \log_a b = \frac{\log b}{\log a} \).

### Step 3: Combine the Terms
When we multiply these fractions, we notice that most terms will cancel out:
\[ \log_{2}3 \cdot \log_{3}4 \cdots \log_{1023}1024 = \frac{\log 3}{\log 2} \cdot \frac{\log 4}{\log 3} \cdots \frac{\log 1024}{\log 1023} \]
This simplifies to:
\[ \frac{\log 1024}{\log 2} \]

**Hint:** Look for cancellation in the product of fractions.

### Step 4: Evaluate \( \log 1024 \)
Since \( 1024 = 2^{10} \), we have:
\[ \log 1024 = \log(2^{10}) = 10 \log 2 \]

**Hint:** Use the property \( \log(a^b) = b \cdot \log a \).

### Step 5: Substitute Back
Now substituting back, we get:
\[ \frac{\log 1024}{\log 2} = \frac{10 \log 2}{\log 2} = 10 \]

### Step 6: Final Calculation
Now we can substitute this back into our logarithm:
\[ \log_{10}(10) = 1 \]

Thus, the final value of the original expression is:
\[ \boxed{1} \]