Proof on logarithms

  • #1
ubergewehr273
140
5

Homework Statement


Prove that ##{1/3} < log_{34} 5 < {1/ 2}##

Homework Equations


##log_b a = {1/ log_a b}##
##logmn = logm + logn##


The Attempt at a Solution


##log_{34} 5 = {1/ log_5 34}##
##= 1/(log_5 17 + log_5 2)##
##=1/(1 + log_5 3 + log_5 2 + something)##
##=1/(1 + log_5 6 + something)##
##=1/(2+something~else)##

something else ##<1##

Hence it is greater than 1/3 and smaller than 1/2.
But I think mathematically speaking, that ##something## isn't supposed to be there.
 

Answers and Replies

  • #2
Fightfish
954
117
A better way to do this is to recast the inequality as
[tex]3 > \log_{5}34 > 2[/tex]
and rewrite ##3## and ##2## in terms of base-5 logarithms as well.
 
  • #3
ubergewehr273
140
5
Thanks. Made it much easier than my solution.
 

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