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

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.