MHB Which is Greater: $\log_9 25$ or $\log_4 9$?

[anemone]

Which of the following two is greater?

$\log_9 25$ and $\log_4 9$

 

[kaliprasad]

Which of the following two is greater?

$\log_9 25$ and $\log_4 9$

Spoiler

$9^{1.5} = 3^3 = 27$
$4^{1.5} = 2^3 = 8$

so $\log_9 27$ = $\log_4 8$
or $\log_9 25$ < $\log_9 27$ = $\log_4 8$ < $\log_4 9$

hence $\log_4 9$ is greater

 

[anemone]

Very good, kaliprasad and thanks for participating!

My solution:

Spoiler

$\log_9 25<\log_9 26<\dfrac{\log_4 26}{\log_4 9}<\dfrac{\log_4 26}{\log_4 8}=\dfrac{\log_4 (26)^2}{3}=\dfrac{\log_4 676}{3}<\dfrac{\log_4 729}{3}=\log_4 9$

 

[Albert1]

Very good, kaliprasad and thanks for participating!

My solution:

Spoiler

$\log_9 25<\log_9 26<\dfrac{\log_4 26}{\log_4 9}<\dfrac{\log_4 26}{\log_4 8}=\dfrac{\log_4 (26)^2}{3}=\dfrac{\log_4 676}{3}<\dfrac{\log_4 729}{3}=\log_4 9$

Spoiler

it should be $\log_9 26=\dfrac{\log_4 26}{\log_4 9}$ (a typo)