Logarithms - Get the second solution to a logarithm equation

Here's the original equation:

$$\ln(11x-10) + \Big(\ln(11x-10)\Big)^2$$ = 6

I've managed to obtain one solution: $$x = \frac{e^2 + 10}{11}$$

through those steps:

1. $$\ln\Big((11x-10)(11x-10)^2\Big) = 6$$

2. $$\ln\Big((11x-10)^3\Big) = 6$$

3. $$(11x-10)^3 = e^6$$

4. $$11x-10 = e^2$$

5. $$11x = e^2+10$$

6. $$x = \dfrac{e^2+10}{11}$$

The textbook shows a second solution: $$x = \dfrac{10e^3 + 1}{11e^3}$$, but how do you get to that result?

• Your original equation has the form $y^2+y=6$, which is a quadratic with two solutions for y. May 18 '12 at 11:44

Your solution is incorrect because :

$\ln(11x-10) +(\ln(11x-10))^2$ is not equal to $\ln(11x-10 \times {(11x-10)}^2)$

To solve

$$\ln(11x-10) + (\ln(11x-10))^2 = 6$$

Let $t=\ln(11x-10)$

$$t^2 +t -6=0$$

this gives , $t=2$ and $t=-3$

when $t=2 \longrightarrow \ln(11x-10)=2 \longrightarrow x=\frac{e^2+10}{11}$

when $t=-3 \longrightarrow \ln(11x-10)=-3 \longrightarrow x=\frac{10e^3+1}{11e^3}$

Hint: Let $a = \ln(11x - 10)$. Then you would have $$a + a^2 =6 \qquad \text{or}\qquad a^2 + a -6 =0.$$