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Contents
raw puzzle

Given the equation 6x+4x=9x6^x + 4^x = 9^x, solve for all xx.

Solution

The idea is to divide the whole equation by 4x4^x:

4x+6x=9x1+6x4x=9x4x1+(32)x=((32)2)x=((32)x)2\begin{array}{rrl} & 4^x + 6^x &= 9^x\\ \Leftrightarrow & 1 + \frac{6^x}{4^x} &= \frac{9^x}{4^x}\\ \Leftrightarrow& 1 + \left(\frac{3}{2}\right)^x &= \left(\left(\frac{3}{2}\right)^2\right)^x = \left(\left(\frac{3}{2}\right)^x\right)^2\\ \end{array}

In this way we get a quadratic formula in q=(32)xq=\left(\frac{3}{2}\right)^x:

1+q=q2q2q1=0(q12)2=54q=12±52q=1±52\begin{array}{rrl} & 1 + q &= q^2\\ \Leftrightarrow &q^2 - q - 1 &= 0\\ \Leftrightarrow &(q-\frac{1}{2})^2 &= \frac{5}{4}\\ \Leftrightarrow &q &= \frac{1}{2}\pm\frac{\sqrt{5}}{2}\\ \Leftrightarrow &q &= \frac{1\pm \sqrt{5}}{2}\\ \end{array}

Since q=(32)x>0q=\left(\frac{3}{2}\right)^x>0, we can ignore the negative solution and proceed with the positive (golden ratio) q=1+52q = \frac{1+ \sqrt{5}}{2}. Setting both variants for qq equal yields

(32)x=1+52xln(32)=ln1+52x=ln1+52ln(32)=1.1868143\begin{array}{rrl} &\left(\frac{3}{2}\right)^x &= \frac{1+ \sqrt{5}}{2}\\ \Leftrightarrow &x\cdot \ln\left(\frac{3}{2}\right) &= \ln\frac{1+ \sqrt{5}}{2}\\ \Leftrightarrow &x &= \frac{\ln\frac{1+ \sqrt{5}}{2}}{\ln\left(\frac{3}{2}\right)} \\ &&= 1.1868143\dots \end{array}

Solution using Newton Raphson method

We can solve this problem also numerically using Newton Raphson method.

Let f(x)=4x+6x9xf(x) = 4^x + 6^x - 9^x, then f(x)=4xln4+6xln69xln9f'(x) = 4^x \ln 4 + 6^x \ln 6 - 9^x \ln 9. Since we want the solution xx, for which f(x)=0f(x) = 0, we try some small values: f(1)=1f(1) = 1, f(2)=29f(2) = -29. That means f(x)=0f(x)=0 must be somewhere between 1 and 2, so we start with x0=1.5x_0=1.5:

The next better estimate for xx is given by xn+1=xnf(xn)f(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}:

xnx_nf(xn)f(x_n)f(xn)f'(x_n)xn+1x_{n+1}
1.50000-4.303062-21.9013301.3035
1.30353-1.105660-11.5607931.2079
1.20789-0.166685-8.2241991.1876
1.18762-0.006124-7.6256051.1868
1.18682-0.000009-7.6025901.1868
1.18681-0.000000-7.6025551.1868
1.186810.000000-7.6025551.1868
1.18681-0.000000-7.6025551.1868