Question

Solve log(x^2-6)=1+log(x-3).

Answer 1

Log(x²-6)=1+log(x-3)
log(x²-6)=log 10 + log(x-3)                  (log 10=1)
log(x²-6)=log(10(x-3))                          (log a+log b=log (a*b))
Then:
x²-6=10(x-3)
x²-6=10x-30
x²-10x+24=0

x=[10⁺₋√(100-96)]/2
=(10⁺₋2)/2

We have two possible solutions:
x₁=(10-2)/2=4

We can check it out this solution:
log(4²-6)=log(16-6)=log 10=1
1+log(x-3)=1+log(4-3)=1 +log 1=1+0=1

This solution is rigth.

x₂=(10+2)/2=6

We can check it out this possible solution:
log(6²-6)=log (36-6)=log 30≈1.477121255
1+log(x-3)=1+log(6-3)=1+ log 3=1.477121255

This solution is right too.


Answer: we have two solutions; x₁=4 and x₂=6.