The equation log(2) * (x) + log(3) * (x) = 0 is solved using the product property of logarithms, resulting in x = ±√(1/6).
Step-by-step explanation:
We are given the equation log(2) * (x) + log(3) * (x) = 0 and are asked to solve for the variable x. This equation can be simplified using logarithm properties such as the product property of logarithms, which states that the logarithm of a product is the sum of the logarithms (log(a*b) = log(a) + log(b)).
Let's apply this property in reverse to combine the logs:
log(2x) + log(3x) = 0
This implies log(2x*3x) = 0, which simplifies to log(6x^2) = 0.
Using the definition of logarithms, we know that if logb(a) = 0, then a must be 1 because b^0 = 1.
Hence, we get 6x^2 = 1. Now we can solve for x:
Divide both sides by 6 to isolate x^2:
x^2 = 1/6
Take the square root of both sides to solve for x:
x = ±√(1/6)
Therefore, the solution to the equation is x = ±√(1/6).