# Proof on logarithms

ubergewehr273

## 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.

Fightfish
A better way to do this is to recast the inequality as
$$3 > \log_{5}34 > 2$$
and rewrite ##3## and ##2## in terms of base-5 logarithms as well.

RUber
ubergewehr273
Thanks. Made it much easier than my solution.