Answer:
1) x=6
2) x=1
3) x=3+sqrt(10)
4) x=2, x=4
Explanation:
All 4 of these problems require using logarithmic identities so I suggest reviewing them and trying this again before looking at the following solutions.
1) Using a logarithmic identity, we can say that log(x)+log(4)=log(4x). Since this is equal to log(24), we know that 4x=24, thus x=6.
2) Since the bases of all 3 log terms are the same, we don't have to be concerned with them. 2log(3)=log(9), and log(9)-log(9)=log(9/9)=log(1), and since log(1)=log(x), x=1. We could also have just said log(9)-log(9)=0 and gone off the knowledge that log(1)=0 and still gotten x=1, but if you don't know this then the first way is more straightforward.
3) Again, the bases are the same so they can be disregarded in this situation. We can convert log(x)+log(x-6) into log((x^2)-6x). Because this is equal to log(1), we can say that (x^2)-6x=1, and use algebra to solve for x. We can complete the square or use the quadratic formula, but we get x=3+sqrt(10).
4) We can change 2log(x) to log(x^2) using another log identity, and we can change log(2)+log(3x-4) into log(6x-8), so we have log(x^2)=log(6x-8). Now, we can cancel out the logs, getting x^2=6x-8. Similar to the 3), we can use algebra to solve for x, getting x=2 and x=4.