+0  
 
0
87
1
avatar

Find \(\large \log_2 3 \times \log_3 4 \times \log_4 5 \times \cdots\times \log_{127} 128 = \, ?\)

 Jun 26, 2020
 #1
avatar+21953 
0

Using the change-of-base formula, writing each of these logs as logs with base 10:

 

log2(3)  =  log(3) / log(2)

log3(4)  =  log(4) / log(3)

log4(5)  =  log(5) / log(4)

...

log126(127)  =  log(127) / log(126)

log127(128)  =  log(128) / log(127)

 

When you multiply all of these together, notice that only  log(3)  from the first factor and  log(128)

from the last factor remain, all the rest cancel out.

 

log(128) / log(2)  =  log( 27 ) / log( 2 ) =  7 · log( 2 ) / log( 2 )  =  7

 Jun 26, 2020

62 Online Users

avatar
avatar
avatar
avatar
avatar