How to solve this equation with $x$ to a logarithmic power?

Question

The equation is $x^4-5x^{\log_25}+9=0$.

Graphic calculators like Desmos show that there are two solutions $x_1=2$ and $x_2≈1.76592$.

So how can I solve my equation to get the value of $x_2$?

Answer 1

Dr. Graubner was right; as frustrating as it may be, there are some simple-looking equations that are impossible to solve with standard algebraic operations. As far as I can tell, this is one of them. Unless some fancy method that I don't know of exists to solve it analytically, I think numerical methods like Newton's method are your best bet.

Answer 2

You could get a good approximation using Taylor series around $x=2$; this would give $$x^4-5x^{\log_2(5)}+9=(x-2) \left(32-\frac{25 \log (5)}{2 \log (2)}\right)+(x-2)^2 \left(24-\frac{25 \log ^2(5)}{8 \log ^2(2)}+\frac{25 \log (5)}{8 \log (2)}\right)+(x-2)^3 \left(8-\frac{25 \log ^3(5)}{48 \log ^3(2)}+\frac{25 \log ^2(5)}{16 \log ^2(2)}-\frac{25 \log (5)}{24 \log (2)}\right)+O\left((x-2)^4\right)$$ Facor $(x-2)$ and solve the quadratic equation in $(x-2)$. The result would be $\approx 1.76469$ while the "exact" solution is $\approx 1.76592$ as you wrote it.