Question 1209834
Let's analyze this expression.

**1. Change of Base Formula**

Recall the change of base formula for logarithms:

log<sub>a</sub>b = log<sub>c</sub>b / log<sub>c</sub>a

We'll use this formula to express all logarithms with a common base, say base 10.

**2. Applying the Change of Base**

* log<sub>4</sub>5 = log 5 / log 4
* log<sub>5</sub>6 = log 6 / log 5
* log<sub>6</sub>7 = log 7 / log 6
* log<sub>2047</sub>2048 = log 2048 / log 2047
* log<sub>2048</sub>2049 = log 2049 / log 2048

**3. Writing the Expression**

The given expression becomes:

(log 5 / log 4) + (log 6 / log 5) + (log 7 / log 6) + ... + (log 2048 / log 2047) + (log 2049 / log 2048)

**4. Pattern Recognition**

Notice that the numerators and denominators form a telescoping pattern. That is, the numerator of each term cancels with the denominator of the next term.

However, the question is missing many elements in the middle. We need to know what the sequence is. Based on the pattern, it is most likely:

log<sub>4</sub>5 + log<sub>5</sub>6 + log<sub>6</sub>7 + ... + log<sub>2047</sub>2048 + log<sub>2048</sub>2049

**5. Telescoping Effect**

Let's assume the missing terms follow the pattern.

If we write the expression as:

(log 5 / log 4) * (log 6 / log 5) * (log 7 / log 6) * ... * (log 2048 / log 2047) * (log 2049 / log 2048)

Then we are multiplying the terms together, and the terms cancel.

However, the question is asking for the sum, not product.

* log<sub>4</sub>5 + log<sub>5</sub>6 + log<sub>6</sub>7 + ... + log<sub>2047</sub>2048 + log<sub>2048</sub>2049

This question is incorrect, the question should be:

log<sub>4</sub>5 * log<sub>5</sub>6 * log<sub>6</sub>7 * ... * log<sub>2047</sub>2048 * log<sub>2048</sub>2049

If the question was a product, then the solution is:

(log 5 / log 4) * (log 6 / log 5) * (log 7 / log 6) * ... * (log 2048 / log 2047) * (log 2049 / log 2048)

= log 2049 / log 4

= log<sub>4</sub> 2049

**6. Correcting the Question**

It is highly likely that the question intended the following:

log<sub>4</sub> 5 * log<sub>5</sub> 6 * log<sub>6</sub> 7 * ... * log<sub>2047</sub> 2048 * log<sub>2048</sub> 2049

**7. Solving the Corrected Question**

Using the change of base formula and the telescoping effect:

(log 5 / log 4) * (log 6 / log 5) * (log 7 / log 6) * ... * (log 2048 / log 2047) * (log 2049 / log 2048) = log 2049 / log 4 = log<sub>4</sub> 2049

**8. Simplifying Further**

Since 2048 = 2^11 and 4 = 2^2, we can say:

log<sub>4</sub> 2049 = log<sub>2^2</sub> 2049 = (1/2)log<sub>2</sub> 2049.

We can't simplify this expression further unless we have additional information about 2049.

**Conclusion**

If the question was a product, the answer is log<sub>4</sub> 2049. If the question was a sum, then the question is likely asked in error, as the sum cannot be simplified in a simple way.