# Math Help - Log comparision question

1. ## 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

2. 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

3. Thanks, but it still is quite difficult to understand.

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