Final answer:
The expression log(2x-3)=log(x+1)+log3 can be simplified by using the property of logarithms that the log of a product equals the sum of the logs. After simplification and solving for x, the apparent solution is x = -6. However, this is an extraneous solution as it leads to a negative logarithm, meaning there is no real number solution.
Step-by-step explanation:
To solve the equation log(2x-3)=log(x+1)+log3, we use the properties of logarithms. Specifically, we use the property that says the logarithm of a product is the sum of the logarithms: log(xy) = log(x) + log(y). With this property, we can combine the right side of the equation:
log(x+1) + log(3) = log[(x+1)*3]
So the equation becomes:
log(2x-3) = log(3(x+1))
Since the logarithms are equal, the arguments must be equal. Therefore, we can set 2x-3 equal to 3(x+1) and solve for x:
2x - 3 = 3x + 3
x = -6
Be careful: the solution x = -6 is extraneous because it would make the argument of the logarithm negative, which is not possible. Hence, there is no solution to the initial equation within the domain of real numbers.