Log comparision question

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January 14th 2009, 05:03 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:

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

Dunno what's next. I can figure out that a<2 and ab>2
January 14th 2009, 05:21 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:

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

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
January 15th 2009, 02:50 PM
Uncle6

Thanks, but it still is quite difficult to understand.
January 15th 2009, 03:49 PM
Jhevon

Quote:

Originally Posted by Uncle6

Thanks, but it still is quite difficult to understand.

where particularly is your difficulty?
January 15th 2009, 06:26 PM
Uncle6

Quote:

in other words, b has the task of increasing the base 5 a larger magnitude (relatively speaking) than a has to do for 3

I get lost here. Would u know how to write it out as a proof?