Question

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

Answer 1

See explanation.

Explanation:

We can solve this equation by frst substituting `log(x-1)` as a new variable:

`t=log(x-1)`

The equation turns into:

`3t^2-10t+3=0`

To solve the equation we use the quadratic formula:

`Delta=b^2-4ac`

`Delta=(-10)^2-4*3*3`

`Delta=100-36=64`

`sqrt(Delta)=8`

`t_1=(-b-sqrt(Delta))/(2a)=(10-8)/6=1/3`

`t_2=(-b+sqrt(Delta))/(2a)=(10+8)/6=3`

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

`1/3=log(x_1-1)=>x_1-1=root(3)(10)=>x_1=1+root(3)(10)`

`3=log(x_2-1)=>x_2-1=1000=>x_2=1001`

Answer:

The equation has `2` solutions:

`x_1=1+root(3)(10)`

`x_2=1001`