let's do some math for fun here we will compare logarithm numbers the first one is log base
2 of 3 and the second one is log base 3 of 5 of course we will not be using a calculator or
WolframAlpha we'll just use our mathematical knowledge hmm let's see what we can do first
of all that may sound observation both of these numbers are positive right because you see that
the input is 3 and then the base is 2 and you can also see that from earlier expression and fountain
to better both of these numbers are bigger than 1 right because if the input was exactly to that log
base 2 of 2 that's exactly what but since the input right here is 3 then this is bigger than
what sings log is an increasing function likewise no this is also bigger than what for the similar
music so let me just make some observation right here I will just write this down we know that both
of these numbers are greater than 1 okay hmm what else can we say though in fact they are both less
than 2 isn't it because we have this is just as 3 if we have log base 2 of 4 that's exactly 2
right but 3 is less than 2 so this is less than 2 likewise if this was an ID then log base 3 of
9 is 2 but since this is a 5 so this whole thing right here will be less than 2 so I was just ready
stuff for you guys both of them are greater than 1 in the meantime they are less than 2 and let me
just do a quick demonstration on this right here ok so suppose if you want to compare log base 3
of 5 Thursday's to okay this is one of the things that you can do you can just take street to this
power and three to this power because maybe the log base 2 is spotting you and you can cancel
this out and you can really just compare 5 verses 3 to the second power is 9 and of course right now
we can say 5 is less than 9 and in fact we can go backwards and conclude that number is less
than 2 because 3 to the X that function is an increasing function all right in other work the
inequality will stay after you apply 3 to this through that the inequality sign will stay and
they can go both way all right so that's the idea hmm so both of them are in between of 1 & 2 and
I'll show you guys how we can compare with the two but now let me just use another number to
compare this with okay and what number is better to try besides 1.5 because we have one already
we have two right here already let's try to do it with 1.5 so let me just put this down I will have
to come here log base 2 of 3 vs 1 point 5 and of course we are all adults now so that's right one
point is 3 over 2 okay and let's just play along with some algebra first of all I see that the log
base 2 is bothering me I can do poverty earlier I can just do 2 to this power and 2 to that power
listen our can so and I have 3 on left hand side and I just have to come here this with that 2 to
the 3 over 2 power unfortunately this exponent right here it's bothering me what can I do well
this is 3 over 2 power it's just that you take the square root to the third power right what
we can do is now take the square on both sides and on the left hand side 3 squared is 9 and we
were comparing this with when you have this to that of course the power you multiply them and
this is just 2 to the third power and we get 8 right and now can we draw a conclusion yes we
can say that 9 yes bigger than 8 now here's the important part can we go backwards yes because x
squared okay it's an increasing function if the input is positive and they are positive numbers
start with actually go backwards right so now I can say three was bigger than 2 to the 3 over 2
power and then we can also go backwards because 2 to the X is an increasing function as well I can
conclude that log base 2 of 3 is bigger than 3 over 2 all right and I'm just applying the
increasing functions for all these steps right here so the inequalities can go both ways right
so that's good and now I will do the same thing with the base 3 right all 5 and I will compare
this with 3 over 2 and let's just go ahead do the same thing this time though I will do 3 to
this power and 3 to that power so that this and now can so and I see that we have the left hand
size 5 I'm comparing down with 3 raised to the 3 over 2 power and then of course similar to earlier
let's square both sides on the left hand side this is 25 on the right hand side this is just 3 to the
third power right which is 27 of course this is 25 this to 27 if you have to compare that it's a
less than like that and we can also go backwards for the simple reason and earlier I took 3 to the
X right well this is still increasing function so I can go backwards like that now take a look
log base 2 of 3 if people and 3 over 2 but log base 3 of 5 it's less than 3 over 2 so finally
I can draw conclusion isn't it so notice let me just put down 3 over 2 in the middle and
the smaller number which is this log base 3 of 5 and the bigger number is that right log base
2 of 3 and of course 3 over 2 is in the middle I don't need to read on three over two so therefore
I can come COO that this it's more than that so log base 3 of 5 it's less than log base 2 of 3
alright so in other word this right here it's actually bigger so did you guys figure this
out did you guys do it this way what else did you guys do teach you guys that's PTO
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