How do I solve this cyclic logarithmic equation?

Question

$$5^{\log_2 x}+2x^{\log_5 2}=15$$ I have also noticed, that logarithmic terms are cyclic and tried to express one as y to make it easier, but still had no luck solving it. Any help?

Answer 1

You can write the equation as

$$x^{\log_25}+2x^{\log_52}=15,$$or with $t=x^{\log_25}$,

$$t+2t^{\log_5^22}=15.$$

There is a single root near $x=2.8988$, with no closed-form expression.

Answer 2

Note that

$$a^{\log_cb}=b^{\log_ca}$$ SO $$5^{\log_2 x}+2x^{\log_5 2}=x^{\log_2 5}+2x^{\log_5 2}=15$$

Let us call $y=\log_25$ So$$x^y+2x^{1/y}=15$$

which has a solution at

$$x\approx2.898813156$$