How to solve $6^{x} + 8^{x} + 9^{x} = 12^{x}$?

Question

I checked on WolframAlpha and it says the answer is approx. 2.56639, but I don't know how to go about solving this equation. What kind of approach should I use?

Answer 1

The equation $$f(x)=6^{x} + 8^{x} + 9^{x} - 12^{x}$$ will not show any analytical solution and cannot be transformed in any polynomial. So, only numerical methods would give the result.

Plotting or using inspection, we know thet the solution is between $2$ and $3$ since $f(2)=37$ and $f(3)=-271$.

The simplest method is probably Newton which, starting from aguess $x_0$, will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ So, let us apply the method starting using $x_0=3$. The iterates will then be $$x_1=2.78153052$$ $$x_2=2.63689146$$ $$x_3=2.57608481$$ $$x_4=2.56659523$$ $$x_5=2.56638730$$ $$x_6=2.56638720$$ which is the solution for nine significant figures.

As you see, the convergence has been rather slow (it would have been faster starting closer to the solution).

But instead of $f(x)$, consider the transform $$g(x)=\log(6^x+8^x+9^x)-\log(12^x)$$ If you plot the new function, it looks that it is almost a straight line and this is much much better. Let us repeat the same calculations starting from $x_0=3$ as before. The iterates are $$x_1=2.56168284$$ $$x_2=2.56638663$$ $$x_3=2.56638720$$ which is the solution for nine significant figures. Much faster, isn't it ?