# Homework Help: Proof on logarithms

Tags:
1. Jun 5, 2016

### ubergewehr273

1. The problem statement, all variables and given/known data
Prove that ${1/3} < log_{34} 5 < {1/ 2}$

2. Relevant equations
$log_b a = {1/ log_a b}$
$logmn = logm + logn$

3. 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.

2. Jun 5, 2016

### 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.

3. Jun 5, 2016

### ubergewehr273

Thanks. Made it much easier than my solution.