Question 1209834: Compute
\log_4 5 + \log_5 6 + \log_6 7+ \log_{2047} 2048 + \log_{2048} 2049
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze this expression.
**1. Change of Base Formula**
Recall the change of base formula for logarithms:
logab = logcb / logca
We'll use this formula to express all logarithms with a common base, say base 10.
**2. Applying the Change of Base**
* log45 = log 5 / log 4
* log56 = log 6 / log 5
* log67 = log 7 / log 6
* log20472048 = log 2048 / log 2047
* log20482049 = 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:
log45 + log56 + log67 + ... + log20472048 + log20482049
**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.
* log45 + log56 + log67 + ... + log20472048 + log20482049
This question is incorrect, the question should be:
log45 * log56 * log67 * ... * log20472048 * log20482049
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
= log4 2049
**6. Correcting the Question**
It is highly likely that the question intended the following:
log4 5 * log5 6 * log6 7 * ... * log2047 2048 * log2048 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 = log4 2049
**8. Simplifying Further**
Since 2048 = 2^11 and 4 = 2^2, we can say:
log4 2049 = log2^2 2049 = (1/2)log2 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 log4 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.
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