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Tuesday, November 19, 2019

2019/019) Solve for positive x 4^x+6^x = 9^x

we have 4=2^2, 6= 2 * 3, 9=3^2
So we get
2^{2x}+( 2 * 3)^x = 3^{2x}
divding by 2^{2x} we get
1 + (\frac{3}{2})^x =   (\frac{3}{2})^{2x}
Putting (\frac{3}{2})^x = y we get
1 + y = y^2
so y = golden ratio \phi = \frac{1+\sqrt{5}}{2}
Giving
(\frac{3}{2})^x = \frac{1+\sqrt{5}}{2}
taking log on both sides we get
x= \frac{\log \frac{1+\sqrt{5}}{2}}{\log \frac{3}{2}}



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