# How do you solve log_(1/2) (x^3 + x) + log_(1/2) (x^4 - 2x) = 1?

Apr 22, 2016

Roots of $2 {x}^{7} + 2 {x}^{5} - 4 {x}^{4} - 4 {x}^{2} - 1 = 0$, with x > 2^(1/3. This has only one real root in (1.29, 1.3). By bisection method , we can get closer to the root. Newton-Raphson method is faster,

#### Explanation:

Using $\log a + \log b = \log a b \mathmr{and} {\log}_{a} a = 1$,.

log_(1/2)(x^3+x)(x^4-2x))=1.

So. $\left({x}^{3} + x\right) \left({x}^{4} - 2 x\right) = \frac{1}{2}$
$f \left(x\right) = 2 {x}^{7} + 2 {x}^{5} - 4 {x}^{4} - 4 {x}^{2} - 1 = 0$

It is easy to prove that this has only one real root that is positive. Also, using sign test, f(1.29) < 0 and f(1.3) > 0. The root is in (129, 1.3.
Really, rounded to 3-sd. it is 1.29.