A Nice Exponential Equation
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- Опубликовано: 6 окт 2024
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I did like this:
3^x * 2^(3x/(x+2) = 3*2
3^(x-1) = 2^(1 - 3x/(x+2)) = 2^((-2x+2)/(x+2)) | log3 both sides
x-1 = ((-2x+2)/x+2) log3_2
(x-1)(x+2) = -2(x-1)log3_2
we can notice here x=1 is solution
then from second possibiity
x + 2= - 2log3_2
so x = -2log3_2 - 2 is another solution
simplifying it's log3_1/36
As usual, we have a simple integer solution, namely x = 1, because 3^1 * 8^(1/3) = 3 * 2 = 6
(The cube root of 8 is 2).
Write 8 as 2³ and 6 as 3.2
Both sides now have same base
Debase both sides and equate exponents
No, 3.2 means 3 and two tenths. Either write 3*2, 3(2), or a multiplication dot between the 3 and 2, for instance.
I got x=1 right away, but I became a little too log/ln crazy for the other solution!
Very nice presentation- thanks a lot 👍
You are most welcome
Guessing method took less than 30 second lol.
Solved by inspection in about 15 seconds.😀
Lol I’m sure you found the irrational solution by inspection.
@@moeberry8226 I'm a rational guy, so not interested in irrational solutions 🙃
Yes same x=1 is the clear solution on first glance
Ah ah... x=1 and the second Solution?
@@mjpottertx lmaooooo good answer bro. You should be a lawyer
Excelente presentación profesor, gracias
Since u are doing some complicated stuffs, I think u are the right person to ask my question
x ^ ( 1/( log x base a) = a , where x > 0 and x not = to 1
Can u explain this for me 😢
Suppose a, x > 0, a 1, and x1. Then, ln(x^(1/ln(x))) = (1/ln(x))(ln(x) = 1. So,
x^(1/ln(x)) = e^1 = e. Thus,
x ^ ( 1/( log(base a) x) = x^(ln(a)/ln(x)) = (x^(1/ln(x)))^ln(a) = e ^ ln(a) = a.
x^(1/logx_a)=a
Take logx on both sides.
1/logx_a=logx_a
Multiply by logx_a on both sides
1=(logx_a)^2
Take squaroot on both sides
±1=logx_a
Use base of change law to get logx_a=lna/lnx
±1=lna/lnx
reciprocal both sides
1/±1=lnx/lna
Since the reciprocal of 1 and -1 are themselves 1/±1=±1
±1=lnx/lna
Use base of change law to get lna/lnx/lna=loga_x
±1=loga_x
Raise both sides to the power of a
To get a^(±1)=x
Hope it helps
@@abhinavanand9032 Your conclusion is that x = a or 1/a. Since this isn't always true, your logic must be flawed. I'm too lazy to find your error(s).
@@someperson188 1/a is a solution
@@someperson188 thank you so much for the explanation. I'm actually a grade 10 student. I accidentally discover this equation when I was doing exercises. Then I wonder how it can give the base value for all value of x. Now I understood. Thanks again.
X=1.
X = 1
Guess and check....got it.
Too tired to work out if it's the only solution.
no
@@SyberMath Haha ok fair enough I need to put some more work in!!! Nice.
Lavorando i dati si ottiene equazioni di 2 grado... x^2+log(3)12x-2log(3)6=0 che dà 2 soluzioni... x=1,x=-3,26185
Nice
Thanks
Actually I challenge you to solve x^7×(1/2)^(x^sqrt6)×(2sqrt6/7)^(7sqrt6/6)=1 there are 2 real solutions, (7sqrt6/3)^(sqrt6/6) or (7sqrt6/6)^(sqrt6/6)
Nice video. I tried to optimise the amount of steps taken in the second method by using a logarithm with base 3 instead of the natural logarithm as all the logarithm tricks work with any base.
Nice work!
N to acustumado com questões cheias de log e in no meio. Ksksksk
x=1
Great video! eline sağlık.
Saolasin ☺️
X=1
x = 1
X=1