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Solve the equation for real values of \(x\)

Solution

We are dealing with an equation with numbers to the power of \(x\). Let’s start by dividing all the terms by \(4^{x}\), we get \[\frac{4^{x}}{4^{x}}+\frac{6^{x}}{4^{x}}=\frac{9^{x}}{4^{x}}\] \[\Rightarrow1+\left(\frac{6}{4}\right)^{x}=\left(\frac{9}{4}\right)^{x}\] \[\Rightarrow1+\left(\frac{3}{2}\right)^{x}=\left(\left(\frac{3}{2}\right)^{2}\right)^{x}\] \[\Rightarrow1+\left(\frac{3}{2}\right)^{x}=\left(\frac{3}{2}\right)^{2x}\] To continue, we make a change of variables. Let \(u=\left(\frac{3}{2}\right)^{x}\) thus \(\left(\frac{3}{2}\right)^{2x}=u^{2}\) \[\] Now after making the changes, we get a quadratic equation of form \[u^{2}=1+u\] \[\Rightarrow u^{2}-u-1=0\] We calculate the discriminant \(\delta \) \[\delta=\left(-1\right)^{2}-4\left(1\right)\left(-1\right)\] \[=1+4\] \[=5\] \[u=\frac{-\left(-1\right)-\sqrt{5}}{2\left(1\right)}\] \[=\frac{1-\sqrt{5}}{2}\] \[\color{red} {Or}\] \[u=\frac{-\left(-1\right)+\sqrt{5}}{2\left(1\right)}\] \[=\frac{1+\sqrt{5}}{2}\] \[u=\left(\frac{3}{2}\right)^{x}>0\] Therefore the solution \(u=\frac{1-\sqrt{5}}{2}\) is to be rejected. \[\Rightarrow u=\frac{1+\sqrt{5}}{2}\] \[\log u=\log\left(\left(\frac{3}{2}\right)^{x}\right)\] \[=x\log\left(\frac{3}{2}\right)\] \[\Rightarrow x=\frac{\log u}{\log\left(\frac{3}{2}\right)}\] \[\Rightarrow x=\frac{\log \left(\frac{1+\sqrt{5}}{2}\right)}{\log\left(\frac{3}{2}\right)}\approx1.19\]
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