Logarithmic Equation

Question

Solve the following logarithmic equation. Be sure to check each possible solution to determine whether or not it is in the domain of both logarithms. Any possible solutions that are not in the domain of both logarithms are extraneous and should be excluded from your solution set.

 

log₄(3x-1)+log₄(x-1)=2

 

 

 

Solution set___________________

Answer 1

log(4)(3x-1)+log(4)(x-1)=2

log(4)[(3x-1)(x-1)]=2

log(4)(3x^2-x-3x+1)=2

log(4)(3x^2-4x+1)=2

4^2=3x^2-4x+1

3x^2-4x+1-16=0 because 4^2=16

3x^2-4x-15=0

(3x+5)(x-3)=0

3x+5=0

3x=-5

x=-5/3 reject

x-3=0

x=3 answer

 

Answer 2

4^{log4(3x-1)+log4(x-1)} = 4²

(3x-1)•(x-1) = 16

 

I'll let you complete the job (involving solving a quadratic equation).  Note domain is determined by these two facts:

x-1 > 0 and 3x - 1 > 0, so don't take any wooden nickels.