Solve for x, given 3 log^2 (x-1) - 10 log (x-1) = -3?

Oct 23, 2017

See explanation.

Explanation:

We can solve this equation by frst substituting $\log \left(x - 1\right)$ as a new variable:

$t = \log \left(x - 1\right)$

The equation turns into:

$3 {t}^{2} - 10 t + 3 = 0$

To solve the equation we use the quadratic formula:

$\sqrt{\Delta} = 8$

${t}_{1} = \frac{- b - \sqrt{\Delta}}{2 a} = \frac{10 - 8}{6} = \frac{1}{3}$

${t}_{2} = \frac{- b + \sqrt{\Delta}}{2 a} = \frac{10 + 8}{6} = 3$

Now we have to calculate the values of $x$ corresponding with the calculated values of $t$:

$\frac{1}{3} = \log \left({x}_{1} - 1\right) \implies {x}_{1} - 1 = \sqrt[3]{10} \implies {x}_{1} = 1 + \sqrt[3]{10}$

$3 = \log \left({x}_{2} - 1\right) \implies {x}_{2} - 1 = 1000 \implies {x}_{2} = 1001$

The equation has $2$ solutions: