LOGARITHM

Algebra Level 2

Evaluate: $(\log _{ 2 }{ 3) } (\log _{ 3 }{ 4) } (\log _{ 4 }{ 5 } )...(\log _{ 2046 }{ 2047) } (\log _{ 2047 }{ 2048) }$

The answer is 11.

3 solutions

Sudoku Subbu
Apr 29, 2015

First we consider $log_{2}3\times log_{3}4\times log_{4}5\times. . . log_{2047}2048$ $=\frac{log 3}{log 2}\times\frac{log 4}{log 3}\times\frac{log 5}{log 4} . . . . \frac{2048}{2047}$ $=\frac{log 2048}{log 2}$ $=log_{2} 2048$ $=log_{2} 2^{11}$ $=11\times log_{2} 2$ $=11\times1=11$

How do you know that $\displaystyle \frac { \log 2048 } { \log 2 } = \log _ { 2 } 2048?$

There should be some proof.

. . - 3 months ago

Ramiel To-ong
Jun 8, 2015

that will result to log2048/log2 = 11

How do know that?

$\displaystyle \log _ { 2 } 3 \times \log _ { 3 } 4 \times \cdots \log _ { 2047 } 2048 = \frac { \log 2048 } { \log 2 } = 11$ .

. . - 3 months ago

Azadali Jivani
Apr 25, 2015

(log3/log2)(log4/log3)(log5/Log4)........(log2047/log2046)(log2048/log2047)
....after cancellation of log3, log4, log5 & so on. only remain as follows
=log2048/log2=11 (Ans.)

There is no proof that $\displaystyle \frac { \log 2048 } { \log 2 } = 11$ .

. . - 3 months ago

LOGARITHM

The answer is 11.

3 solutions

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