How to solve 9^x -6^x =4^x |-Exponential Equation |-Algebra problem (@mathsjourney6608)

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[@glupilazni1399]

Much simple solution is no matter of X:
9 will always give you number ending with 1 or 9,
6 will always give you number that ends on 6,
4 will always give you number that ends with 4 or 6
In no matter which real number combination you will never get the number that ends with 4 or 6

[@user-sd9bo2hi6z]

1.186

[@Xephoriath]

Amazing explanation and approach, I figured I'd give the problem a try before watching and came to this conclusion, numerically the answers are the same but slightly different approach/outcome. Thanks for sharing this fun problem!
9^x=6^x+4^x
ln(9^x)=ln(6^x+4^x)
xln(9)=ln((6^x)*(1+(4/6)^x))
xln(9)=xln(6)+ln(1+(2/3)^x))
xln(9)-xln(6)=ln(1+(2/3)^x))
xln(3/2)=ln(1+(2/3)^x))
e^(xln(3/2)=e^(ln(1+(2/3)^x))
e^(ln((3/2)^x))=e^(ln(1+(2/3)^x))
(3/2)^x=1+(2/3)^x
(3/2)^x=1+(3/2)^-x
u-substitutiuon:
u=(3/2)^x
u=1+(1/u)
u^2=u+1
u^2-u-1=0
Quadratic Formula:
u=(-(-1)±sqrt((-1)^2-4*1*(-1)))/(2*1)
u=(1±sqrt(5))/2
Plugging u back in:
(3/2)^x=(1±sqrt(5))/2
xln(3/2)=ln((1±sqrt(5))/2)
x=ln((1-sqrt(5))/2)/ln(3/2)
x=Not a Real Number
x=ln((1+sqrt(5))/2)/ln(3/2)
x≈1.186814 Final Answer!

[@mathsjourney6608]

Great.

[@piotrsobodzian5232]

The closed form solution is: x = asinh(-0.5)/ln(2/3) and x = -asinh(-0.5)/ln(2/3)

[@ChristalinePosiyano-nb5mg]

Hy...why did you chose to devide throughout by 3^2x not by 2^2x?

[@mathsjourney6608]

Thanks , because we want the true solution.

[@rick57hart]

1.186 814 390 280 982

[@samwinchester2094]

you can solve it more easier if you divide each number to 4^x

[@ichiyt3762]

why we didnt divide on 2^x as well

[@emidiocavucci884]

For me the calculator reports x= 4.879 considering the "+" in front of the root

[@TheHorrorsPersistButSoDoI]

How can something positive ^x be negative?

[@mathsjourney6608]

Thanks for watching, please can you highlights where you talking about?

[@TheHorrorsPersistButSoDoI]

​@@mathsjourney6608the soultion at the end, you should have added u=(-1-sqrt5)/2 is not possiblle