# Log comparision question

• Jan 14th 2009, 06:03 PM
Uncle6
Log comparision question
"Which is greater, $\log_3{5} or \log_5{11}$? Explain"

Although u can plug and chug into your calculator, this question seems like it's meant to be solved analytically.

I got to:

$\log_3{5}=a , \log_5{11}=b$
$3^a=5, 5^b=11$
$(3^a)^b=11
$

Dunno what's next. I can figure out that a<2 and ab>2
• Jan 14th 2009, 06:21 PM
Jhevon
Quote:

Originally Posted by Uncle6
"Which is greater, $\log_3{5} or \log_5{11}$? Explain"

Although u can plug and chug into your calculator, this question seems like it's meant to be solved analytically.

I got to:

$\log_3{5}=a , \log_5{11}=b$
$3^a=5, 5^b=11$
$(3^a)^b=11
$

Dunno what's next. I can figure out that a<2 and ab>2

think of it this way:

we have $3^a = 5$ and $5^b = 11$

Note that 5 is less than twice 3, while 11 is more than twice 5. in other words, $b$ has the task of increasing the base 5 a larger magnitude (relatively speaking) than $a$ has to do for 3. $b$ is larger

this is a more intuitive approach. which i suppose is the level of thinking this problem requires, since it is elementary/middle school math. there are more rigorous proofs, i'm sure
• Jan 15th 2009, 03:50 PM
Uncle6
Thanks, but it still is quite difficult to understand.
• Jan 15th 2009, 04:49 PM
Jhevon
Quote:

Originally Posted by Uncle6
Thanks, but it still is quite difficult to understand.