Indian Olympiad Mathematics | Exponential Equation

[bollyfan]

Use log base 4 on both equations

2 x log4(5) = 3 log4(3)

3 y log4(3) = 2
3 log4(3) = 2 / y

2 x log4(5) = 2 / y
log4(5) = 1 / (xy)
4^(log4(5)) = 4^(1/(xy))
4^(1/(xy)) = 5

[think_logically_]

From 2nd equation: 4^2 = 3^(3y) -> 4^(2/y} = 3^3 -> 4^(1/y) = 3^(3/2) .
From 1st equation: 3^3 = 5^(2x) -> 3^(3/2) = 5^x.
Substituting expression for 3^(3/2) from previous line into the starting line gives: 4^(1/y) = 5^x, therefore 4^(1/xy) = 5.
No taking logarithms was needed!

[nitingl]

Even though Second Method is Simple, we must understand First Method as it gives Comprehensive understanding of log and exponential functions correctly.

[isiakuisa]

May the almighty Allah give you more wisdom to continue educating us, amin.

[joaobrigido]

Lindíssima questão! PARABÉNS E FELICIDADES!

[Stig_JR]

The second method was wonderful ❤️
Thank you

[liamgauvreau]

I did it in almost a minute by just expanding the given equations to get 25^x=27 and 27^y=16. Using the first equation, raise both sides to the power of y to get 25^xy=27^y=16, so we now have that 25^xy=16. You can check that neither x=0 nor y=0 are solutions, so the number 1/xy is perfectly well defined. Thus, you can raise both sides of the equation to the power of 1/xy to get that 25=16^1/xy. From here we have that 25=4^2/xy, so just square root both sides and you'll get that 5=4^1/xy.

[owlsmath]

Great problem and explanation! I used the 2nd method but the 1st method gives a very reliable approach.

[SarthakThukralst]

Bro you are really good at mathematics... i wish to become like you 🤞🤞

[nanaamoakoh]

I'm in love with the second method 😊

[manafro]

Thank you for your quality content! Content like this makes the Internet such a great place! :) You're awesome!

[adaramolabunmi]

Wonderful. May God continue to increase your wisdom.

Please what is name of this device you are using in teaching.

[mariomestre]

Genial. Activa los subtitulos, merci

[ebuleprecious]

Logarithm rules are more complex than I thought

[lazaremoanang]

So simple 4^(1/xy)=5. I see the time 9:56, if I have the time to watch it, I'll see why it's too long.

[najibamaro]

The second method is good, glad to watch until it ends 😆

[emrealbayrak]

Hi from Turkey.Thank you for this good question.

[naveenprakash]

Please logritham chapter teach us start this lesson