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

1 Answer
Oct 23, 2017

Answer:

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#