Cancelling a logarithm

Question

I was wondering if there was a way to cancel out a logarithm? For example:

$\log_a A$ > $\log_a B$

What would a have to be for the log to go away and be left with A > B? Thanks in advance!

smingerson · Answer 1 · 2016-04-30T13:20:47.003

1

As long as $A, B >0$, if $\log_a(A)>log_a(B)$, then $A>B$ since logarithmic functions are $increasing$ as long as $a>1$, i.e. if $a>b$, $\log(a)>\log(b)$

$A, B>0$ is necessary since otherwise $\log(A)$ (or B) isn't defined.

edited Apr 30 '16 at 13:20

answered Apr 30 '16 at 13:19

smingerson

431

@Joanpemo Thanks for the correction, didn't think through that quite enough. – smingerson Apr 30 '16 at 13:21

score 0 · Accepted Answer · answered Apr 30 '16 at 13:21

0

Assuming everything is positive here, and $a\ne1$, the answer is:

If $a>1$, then $\log_a b <\log_ac\iff b<c$
If $0<a<1$, then $\log_ab<\log_ac\iff c<b$

answered Apr 30 '16 at 13:21

Ty for the answer and correction! ;) – Alex Apr 30 '16 at 13:34

Cancelling a logarithm

2 Answers2