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Find all real x that satisfy 9^x - 4*3^x - 6 = 0
 

 May 24, 2022
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\(9^x - 4*3^x - 6 = 0\\ 3^{2x} - 4*3^x - 6 = 0\\ let \;\;y=3^x\\ y^2-4y-6=0\\ y=\frac{4\pm\sqrt{16+24}}{2}\\ y=\frac{4\pm\sqrt{4*10}}{2}\\ y=2\pm\sqrt{10}\\ 3^x=2+\sqrt{10}\qquad or \qquad 3^x=2-\sqrt{10}\\ \text{But 3 to the power of any number must be positive so}\\ 3^x=2+\sqrt{10}\\ log_33^x=log_3(2+\sqrt{10})\\ x=\frac{log_{10}(2+\sqrt{10})}{log_{10}3}\\ x\approx 1.4940 \)

 

I will leave you to check it.

 

 

 

LaTex:

9^x - 4*3^x - 6 = 0\\
3^{2x} - 4*3^x - 6 = 0\\
let \;\;y=3^x\\
y^2-4y-6=0\\
y=\frac{4\pm\sqrt{16+24}}{2}\\
y=\frac{4\pm\sqrt{4*10}}{2}\\
y=2\pm\sqrt{10}\\
3^x=2+\sqrt{10}\qquad or \qquad 3^x=2-\sqrt{10}\\
\text{But 3 to the power of any number must be positive so}\\
3^x=2+\sqrt{10}\\
log_33^x=log_3(2+\sqrt{10})\\
x=\frac{log_{10}(2+\sqrt{10})}{log_{10}3}\\
x\approx 1.4940
 

 May 26, 2022

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