Mexico - A Nice Math Olympiad Exponential Problem
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- Опубликовано: 15 июн 2023
- Maths Olympiads are held all around the world to recognise students who excel in maths. The test is offered at many grade levels and provides them with numerous possibilities to win certifications, awards, and even scholarships for higher studies.
At the end you can simply apply the definition of logarithm...
2^x = 5 ---> x is the power to wich the base 2 must be raised in order to obtain 5
so you can write:
x = log2(5)
No, she has to do a 5-minute thesis on it (to go from 2^x=5 to x=log5 to base 2, otherwise her students might not understand it.
@@tintiniitkwe learned it as logba=e where e is the exponent b is the base and a the “answer”. its very helpful
I feel the quick way to approach this problem is to recognise that we have a polynomial where both y^3 and y exist. This should ring a bell that we should find some cubed number somewhere. We can write 130 = 125 + 5 = 5^3 + 5. That's it. Immediately we get
y^3 - 5^3 + y - 5 = 0, or (y -5)(y^2 + 5y + 26) = 0, where y = 2^x.
Here onwards, it is simple.
(8×15)+(2×5)=130
thats what i did, i got hung up because i did log(5)-log(2) instead of log2(5)
Wow. This was way easier! I should've just thought through some numbers and would've hit it just by thinking of those 1st few perfect cubes.
y=2x , find the value idiot
She made it way harder to solve than she had to.
This is why I'm going back to watching cooking videos.
Or funny cats
Come on. Everyone should know how to solve this
@@Aman-nk5uq he means it's too easy
Most mathematicians probably can't cook worth a damn. Most people can't.
😂😂 I don't blame you
130=26×5=(5^2+1)×5=5^3+5;
8^x+2^x=(2^x)^3+(2^x);
2^x=5
My high school math teacher used to tell me,to understand an equation better you need to dis-simplify the brief side other than to simplify the complex side.
Wowwww😮😮😮
Nice! I did it like this as well or, similar.
Yeah, it just depends. A lot of times that advice will work, but not always.
Yours can't guarantee there is unique root.
@@zhenyuzhai4098 I know it should be unique in this case
Me given IIT 20 yrs back and working in IT from 15+ yrs still willing to learn this so that I can teach my kid. Feels like back to square one. 😂 happy learning guys.
Looks like you're stuck in a loop in life.
Don't teach to kids whatever that's not useful. Times are changing rapidly. Pretty sure you never used this in your working life.
*for 15 years
@@krishmavthese type of outside the box question are what asked in sof olympiads
@@ankurmondal3220so you are studying fkr sof olympiads, go and give the real IMO
Make sure you don't piss in your pants
Instead of using decimals you could have used more logarithmic identities when you were checking. I think that would have been cleaner.
Came here to note that.
Exactly.. 2^(log 125/log 2) = 125 and so on.
@@manojpadmanabhan2615 let's be nice with our math mates
FASTER APPROACH : 8^x+2^x is always increasing function, hence one root only. Setting x equal to some numbers, we realize that x is between 2 and 3. Consequently,y is between 4 and 8, but y is obviously less than 6 by analyzing cubic equation. So y is between 4 and 6, try 5 and we get that y=5. Everything can be solved in mind.
I did it like this...
2^(3x)+2^x=130
Let 2^x=y
Then y³+y=130
By observation, y=5
Hence
2^x=5
X=log(2)5, ie log 5 base 2
Overall, this problem was on the easier side if it's from olympiad, as I and my friends (Indian high-school students) were able to solve it pretty easily 😅
Hello.
I am a mathematics enthusiast from Nigeria and I really want to increase my knowledge in mathematics. I would like to connect with you.
Do you mind?
Same thoughts here. I literally solved it in mind using the similar way as you in 1min
Yeah but try to solve it when u were 13 or 15.
Damn we really dont need a 8 minute to solve this one. Nice observation.
people @13-15 in india can solve this way faster compared to the older folks in india btw@@m3zuss
Change of bases can be used to simplify the final expression
8^log_2(5) = (2^log_2(5))^3 by associativity and commutativity
=5^3 since x^log_x(a) = a
=125
And the same with the other term, so
125+5 = 130
I agree, the finish could have been done a bit more elegantly with out finding the decimal approx.
Incredible how much effort you put in this. 130=5³+5¹ and this is identical to your y³+y. Therefore y = 2^x = 5. => x = log2 (5). Easy.
It’s just to prove that there is no other solutions with real values
You also have to check for other possible solutions. Duh!
y3+y is stricly increasing, thus has only 1 real root, no need to complexify all what he did, like he did with all his base 2 and 5 logarithms.
Just write 2^x=5 equals to x = ln(5)/ln(2), it's the basics of logs and exponentials
Just subtract 130 from both sides and solve for the x-intercepts of the equation 8^x + 2^x - 130 = y. When y= 0 the graph intercepts the x axis.
I loved my Math's Teacher Sir A Hameed Wayn, in Metric Class ... And after him, you are the 2nd one, whom I would like to praise ..
Since my First Teacher changed the School , and just for Mathematics I followed him to that school. From there, you can see my love for Mathematics . Liked your V-Log ... Though i dont know your name .
I remember fondly a time where maybe, just maybe, I might have had some idea, but I haven’t used anything beyond grade 9 algebra in 20 years.
All these problems seem a little too easy for olympiads
Are you sure?
@@estefanocrespo7930yes these are like middle school problems for us
@@estefanocrespo7930im not him but yeah this is pretty easy for olympiads because i can even answer it
Maybe it's the special olypiad
😂😂😂😂
We will also have complex values of x...
Since we have a cubic polynomial y^3+y-130=0, it will therefore have 3 roots out of which 1 is real (log5/log2) and other two are complex
Other two values of x hence will be: x=log(5+- isqrt71)/2/log2
The imaginary solutions were extraneous. That's why they were explicitly discarded.
@@NicholasOfAutrecourtextraneous or extra anus?
For all functions of type f(x)=ax^3+bx+c the equation f(x)=0 can have only one real root in case of a>0, b>0, because f'(x)=3ax^2+b>0 and therefore f(x) is monotonically increasing. Also, if the equation f(x)=0 has rational roots in the form p/q, p is a divisor of c and q is a divisor of a. If a=1, the rational roots are always whole numbers. One can immediately see that in the specific example f(y)=y^3+y-130=0 , y=5 is a solution, because we first check the whole divisors of 130 (+-1, +-2, +-5, +-10, +-13).
Try to solve x^7+x^5+x^3+x=170 with your method....No chance. With above, you show that x=2 is the only root in just 2 rows.
This is why I know anatomy so well!
So hard. Congrats. Several math concepts 👏👏👏👏👏
First observe that 8^x=(2^3)^x=(2^x)^3. Then set 2^x=y and you get the equation: y^3+y=y*(y^2+1)=130. A few trials gives y=5 and thus x=ln(5)/ln(2)=log(5)/log(2).
Your handwriting and logical thinking ability are awesome ❤
Thanks you so much for refreshing my memory from 10 years ago, Now I wish I have continued on the math field instead
For 2^x=y
y³+y=130
y(y²+1)=130
Clearly for y=5, equation satisfied so x=log5(base 2) is the correct answer.
Having got to y³+ y = 130 it's not hard to try a few small numbers and see that y=5 is a solution. From that the quadratic part could be worked out, though it should be obvious that there can be no other real solutions since y>5 would be too big and y
Yes I think so
Два графика функций
у=х^3 и у=-х+130 пересекаются в одной точке в 1 четверти=> х=5 единственное решение
To all the pretentious people who keep commenting that she made it hard, please note that she is trying to teach everyone at any level, that's why she chose the most straightforward method that any type of student can follow. She is not teaching only the undiscovered geniuses such as yourselves.
very informative and easy way of teaching
Let a=2^x, then a^3+a=130, so a=5 is a solution, then x= ln5/ln2. The other solutions can be found by factoring out (a-5) and solving the quadratic.
The longest way to do that equation.
It's too complicated, I mean after getting 2^x = 5 we get in accordance with the definition of a logarithm x = log2(5) where log2 is a logrithm with base 2. All following calculations
in the video are unnecessary.
Yeah, exactly my thoughts: just use the definition!
It was painful to see how she derives x after getting 2^x = 5 (((
4:50
Redundant. By definition of logarithm, it is the value of the exponent to put on the base to get the argument of the logarithm. So basically in this case x is essentially, in base 2, log(5).
Great session
I think 5 is a rather easy to find "obvious" solution by searching a integer which cube is close to 130. Then it becomes a polynom division.
The proposed factorisation is nevertheless very smart !
In solution checking
a^(log c to base a) can be written as c^(log a to base a) .
So 8^(log 5 to base 2) can be written as 5^(log 8 to bas 2), which is 5^3 and 2^(log 5 to base 2), is 5^(log 2 to base 2) which is 5
5^3+5=130
Really crazy how easy this problem would be for my past self studying engineering. Being done with school and doing the same shit over and over again at work really rots your brain
Это не легкая задача
Challenge your brain in your free time or find a more interesting job (when possible) 😊
@@Peter-Alexander yeah man I'm learning an additional 2 languages right now and also learning more music theory. Trading is also my side thing so I think the analytical part of my brain is still working to some extent. It's just that complex math isn't really my thing these days
If you are asked for only the Real solution then youbare correct. If you want all solutions, you must take the quadratic into account and get two more Complex solutions.
Don't be clever. When is a maths solution going to ask for complex solutions
@@planomathandscienceanywhere above high school, and at high school levels in some countries, expect a full answer unless otherwise specified.
yeah! I want all answers!
reminder me my school years in one of the best math schools in Russia 25 years ago, now all forgotten but still these problems are solvable on the fly almost🙂 good times it was
Какая школа?
y(y²+1) = 5(5²+1)
...at this stage it should be apparent that y = 5 🤪
Given y = 2^x
Then 2^x = 5
log²(2^x) = log²(5)
x = log²(5)
I do not yet have the mathematical experience to have come up with that line of thinking to jump to thinking of the factorisation of 130 then rewriting as factor by grouping. I got to the step where I substituted u = 2^x but had no idea that was actually the way to proceed. I was stuck at what to do now with the 130 as I had no obvious way to factorise u^3 + u - 130 = 0.
How does this relate to real world practicality, thank you.
Before watching:
This is not one that can be easily solved by simply plugging in integers and hoping for a result. Our solution is going to be somewhere between 2 (8^2 + 2^2 = 64+4=68) and 3 (8^3 + 2^3 = 512+8 = 520), and probably closer to 2 than to 3.
Therefore, we have to actually do some calculations.
Alright, 8 = 2^3, and (a^b)^c = a^(bc). Thus 8^x = (2^3)^x = 2^(3x).
We declare U = 2^x. Then 8^x = 2^(3x) = u^3.
Then we have u^3 + u = 130.
Subtract 130 from both sides to get u^3+u-130=0.
Now, we attempt to factor this. The factors of 130 are 1, 2, 5, 10, and 13.
Of those, U=10 gives us results far too large, and U=2 gives us ones far too small. U=5 gives us 125+5-130 = 0, which is accurate.
Thus, (u-5) is a factor. after doing some division, we can factor the equation into (u-5)(u^2 + 5u + 26 )=0.
Going to use the quadratic formula on the second factor, we note that the discriminant is negative. Thus, these are not real roots, so we can skip this section.
Thus, we will go with U=5. However, we're not done! That's the solution for U, not for X.
U = 2^x. Thus, we have 2^x = 5.
Take log_2 of both sides:
Log_2(2^x) = log_2(5) -> X = log_2(5)
Log_2 of 5 will be between log_2 of 4 (2) and log_2 of 8 (3), likely closer to 2 than 3. This checks out.
If you want to change this to a different log using change of base, you may do so. log_b(a) = (log_c(a))/(log_c(b)),.
Then using natural log ln, with a =5 and b = 2:
x = (ln 5)/(ln2).
(We're not doing this for the sake of precision, but rather so it can be easily checked on a calculator. A lot of calculators don't have functions built into them for logs of different bases, at least not ones that you can get to easily. Thus, you have to switch to either common log (log_10) or natural log (log_e))
Superb
☠️
Aah my favourite maths algebra those days ❤🎉
Ótima explicação. Mas, quando você já havia encontrado que 2 elevado a x era igual a 5 , já poderia ter usado a definição de logaritmos e chegar direto na conclusão que x é igual a log de 5 na base 2 .
Ou fazer assim seria um erro matemático?
Parabéns pelo excelente vídeo!
I also wondered why we need last manipulations with Log, it is redundant. 2^X=Y. X = logY.
Beautiful!
Your voice is so soothing
This is hard core algebra. Cubic equations, quadratic equations w/ the quadratic formula, logarithms, fractional exponents, ect…. I did stuff close to this level in high school. The difference was that it did not have multiple layers of this complexity. My takeaway from that experience was that algebra wasn’t hard so long as you worked a lot of different problems and got plenty of practice.
if only the olympiad problems were this nice, solved in 2 seconds
To complicate solution
We can divide by 2^x and y= 2^×
Confuse the number 26y - 25y, can you explaine it!
Use substitution. Substitute 2^x =y and solve polynomial equation.
Use rational zero theorem.
Y=5.
And divide the function by (y-5) and get the quadratic.
Much faster....
I have simplier solution. Multiply both by root x, so then 8+2 = x base root of 130. Which easily = 2.3....
2^log²5 = 5 (by definition)
8^log²5 = (2^3)^log²5=
(2^log²5)^3=
5^3=125
There is no need to do approximete calculations.
In Italy this is called "col senno di poi"! 😂
Gives me shivers, brings back nightmarish memories of struggling with all this in school years!😮
But enjoying it too, in a contradictory way, because there is no pressure!😊
Go easier by using Factorizing with polynomial and basic logarithm
Time to use synthetic division 😁
This is an imperfect solution. I realized that the power I'd 8 could not be 3 abs 2 was too small, meaning it was not going g to be an integer. Kind of ridiculous in my book.
Thank you 🥰🎉👏
great video but the verification can be a little bit better if it's like this:
8^log5 base 2 + 2^log5 base 2 = (2^3) ^log5 base 2 + 2^log5 base 2
= (2^log5 base 2) ^3 + 2^log5 base 2
as we already know a ^logN base a = N
=> (2^log5 base 2) ^3 + 2^log5 base 2 = 5 ^3 + 5
= 125 + 5
= 130
excellent video keep it up and upload more videos like this.
Wawoooo. So simply explained
This can be solved by vanishing factor method
By putting y=5
y-5 is a factor
5^3+5-130=0
y^2(y-5)+5y(y-5)+26(y-5)=0
(y-5)(y^2+5y+26)=0
y-5=0 or y^2+5y+26=0
y=5.
D for quadratic equation
5^2-4×1×26 is less than 0
No real roots.
Solution will be y=5
Although I knew the answer to the problem, being lazy I preferred "Hit and Trial Method" from the given option for these type of questions. This saves a lot of time 😅😅
Waiting for more videos, Mam
Спасибо! Не понимаю концовку с таймкода 4:38, и так ясно, что логорифм - это степень числа по основанию.🧐
Yeah I got it too. But when you check it, you should not convert to log value and you can simply find it.
I could have just put some value in place and solved mathematically to make a guess that i have to go up or down a little. Than three attempts later i might have got x=2.3
I have done it many times. You dont need to go in multiple algebrac complications to solve a problem this simple. Afterall numbers are used to avoid hit and trial of actual objects. We can use numbers for hit and trial as that wont even use extra ink or time cz the equations are more complex than hit and trial. Smart work is also intelligence!
I looked at this for like 2 minutes without a thought of any complex maths and thought the answer might be X= 2.25 and I’m honestly pretty pleased. Lol
I also have zero training in mathematics outside of high school 10+ years ago so go very easy on me
Nicely done. Weird how your 2's are written differently even in the same equation
I think the easiest solution is that we know 2^7=128 next is 8^(1/3) = third root 8 =2 so 128+2=130
This was my solution as well. Much simpler.
Both the exponents are “x” so the value needs to be same. Can’t be 7 for one x and 1/3 for the other
Love this, Thanks ❤
😊😊😊Vivian
Lhu 0:14 0:14
I love these, I do data analysis for word and they have no practical application but I still love them
i solved it like this after 2^x(2^2x+1) = 130 so taking factor of 130 as two terms like 65X2 or 26X5 or 13X10 and trynna see which of these two terms satisfy that eq which is on taking 2^x= 5 so thats the answer after taking log of this
Nice
Instead of that lengthy solution, let 2^x=y, and then let f(y)=y^3+y-130
For y=5, f(5)=0
f'(y)=3y^2+1, which is positive for all values of y, meaning f(y) is a monotonically increasing function, which makes y=5 the only root of f(y)
Then 2^x=5 and solve using logarithmic properties
Yeah, but this is only shorter because it happened to be monotonically increasing. If it wasn’t, you would’ve done this step for no reason, and still had to do the lengthy solution.
I think the question is ill phrased because the types of the symbols in the equation are not specified. For example: no solution over integers. One solution over reals. 3 overvcomplex numbers. But what if we consider prime fields F_p with p>130 or any other algebra type where those symbols could be interpreted in?
I think you could simply write 2^x = 5 => x = log2(5) without all these log divisions, because log2(5) literally means power to which we have to rase 2 in order to get 5, which is x in our case
In real exams, you can just substitute the given choices if satisfies the 130. It will save lot of time instead of solving.
In real exams you are not given choices. What educational system did you attend? "Fast food and exams Inc."?
@@AceGunner72 maybe you didnt take any licensure exams.
Lol, I've done that independently from RUclips thumbnail and my result was X = 1 / [ log130(10) ] which is X = log10(130) which reads as [ log 130 to base 10 ]
Edit: It was quite close as my result evaluates to 2.1139 and the actual result evaluates to 2.3219
I learn my level is olimpic thanks to this channel.😂
은가은씨 오늘도 즐거운 불금 되세요.....
well explained.
how to complicate simple stuff
I got from a much easier and faster way 2*3x and 2*x should be equal to 130 so the sum of two of 2 power something should be equal to 130 which is close to 2*7 hence if x=2 the answer is 64+4 and if x= 3 it would be 528, therefore, x should be between 2 and 3 and closer to 2 than 3 and 3x kinda feels be close to 7 as 128 is close to 130 so 7/3=2.33 and as it wouldn't add up (cuz 127+4,9 is greater than 130 we should decrease 2,33 by 0,01 per time to find the right answer then we got the answer of 2.32
I am a university student in Turkey. We were learning this in primary education. It is a very simple question. Frankly, I was surprised.
Di mi yaa bir de çözümü çok uzatmış biz otomatik olarak logaritma formülüne çakıp geciyorduk 😂
This is a very easy question for high school students in Vietnam
In this particular case easier to resolve as: y(y^2+1)= 130.
130 = either 13 x10 or 26x5.
Then, y= 5, only one solution
Let y=2^x then y³+y-130=0
Easy to see y=5 is one solution. I will skip complex solutions which just requires synthetic division and quadratic formula to solve.
5=2^x -> log_2(5)= x -> x= ln5/ln2
We can do it faster.
130 has only 3 non repeating prime factors 5, 13 and 2. So there is not much of a substitution job to check which one satisfies
Y * (Y^2 +1) = 130
And its 5.
Why aren't the imaginary solutions taken into account? I mean, they are solutions to the original equation, aren't they? Is there an assumption that we must find the real solutions only?
..am I the only one to solve it in their mind after seeing the thumbnail?
yes. You're the only one.
Thanks
Its a mid level math problem for college exam in Turkey.
3/4, very simple
Log 2^x = x log 2. Please learn this. For your solution:
X= log5/log2.
Beautiful question. It's answer is log5/log2 both on the base 10 👌👌
or any other base for that matter
Why aren't we considering the non-real solutions? They're pretty straightforward too!
1st: x=log2(5)
2nd: x=log2(-5/2 - 1/2 (i sqrt(79)))
3rd: x=log2(-5/2 + 1/2 (i sqrt(79)))
Pretty easy!!!
I did it lot quicker using logaritm.
Take log on both sides
Log(8^x + 2^x ) = log(130)
Log 2^x (2^2 + 1 ) = log(130)
X log 10 = 2.113943
X = 2.113493
Lot quicker
Could you explain what steps you took to simplify the left side?
@@MichelangeloPartipilo
Log(8^x + 2^x ) = log(130)
Log 2^x (2^2 + 1 ) = log(130)
X log 10 = 2.113943 (Since log 10 base 10 = 1, hence X was multiplied by 1 leading to final answer)
X = 2.113493
this is extremely easy for an olympiad question
I literally solved it in my head, this is too easy to be an olympiad question
As soon as she got it to Y Cubed plus Y = 130, the answer jumped out at me and I yelled it out loud, trying to beat her to the punch. Who knew I actuality had time to go check the mail, first?
She is not solving math problem she is making math problem.
Was in it better to approximate x between 2 and 3 at a glance?
i wrote all the a,b,c,d's & 1, 2,3,4's at the top and at the end i just put "hence proved" i got 2 marks :) - back benchers ;)