3

I need to know how i should compare logarithms with different bases

Eg:

  • $\log_4 1/15$

  • $\log_3 (1/2)$

  • $\log_5(1/30)$

Witch is greater? I need valid reasoning and proof if possible! Thanks.

jonsno
  • 7,521
shadi
  • 35
  • Convert them to a common base.. then note that logarithm function is increasing if base $> 1$ – jonsno Apr 11 '17 at 11:45
  • how can i convert bases 4,3 and 5 to a common base? – shadi Apr 11 '17 at 11:48
  • Use base converter formula – jonsno Apr 11 '17 at 11:49
  • You can put the examples that you gave in order by approximating their values. For example, 15 lies between 4 and 16, so log 15 to base 4 lies between 1 and 2, so log 1/15 to base 4 lies between -1 and -2 etc. – gandalf61 Apr 11 '17 at 11:50
  • that can work but if numbers were very close together .this might not help right?and i have to convert bases ?is there any other way to compare these kind of logarithms? – shadi Apr 11 '17 at 11:54

1 Answers1

2

Play around with the values. Estimate them. See what happens. For example:

Let's see... $\log_4 (1/15)$.... hmmm.... well, $\log_4(1/16) = -2$... and $\log_4(1/4)=-1$.... so, $\log_4(1/15)$ is between $-2$ and $-1$.... okay, that's pretty good.

How about this one? $\log_3(1/2)$... hmmm..... $\log_3(1/3)=-1$,... and, hey I see it... $\log_3(1)=0$, so $\log_3(1/2)$ is between $-1$ and $0$, so $$\log_4(1/15) < -1 < \log_3(1/2) $$

Lee Mosher
  • 120,280