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.