Final answer:
The logarithmic equation 2log_5 (sqrt(x)) - log_5 (6x-1) = 0 simplifies to log_5 (x/(6x-1)) = 0. Further simplification using logarithmic properties leads to the quadratic equation 5x-1 = 0, which has a solution of x = 1/5. Verification of the solution within the domain of the original logarithms is recommended.
Step-by-step explanation:
To solve the logarithmic equation 2log_5 (sqrt(x)) − log_5 (6x−1) = 0, we first utilize the logarithmic property that allows us to combine logs by division when they are subtracted: log_a - log_b = log(a/b). Using this property, we can rewrite the given equation as:
log_5 ((sqrt(x))^2/(6x−1)) = 0
We know that (sqrt(x))^2 = x, thus simplifying the expression inside the log:
log_5 (x/(6x−1)) = 0
Using the property that log_b(a) = c means b^c = a, we can rewrite this equation as:
5^0 = x/(6x−1)
As any number raised to the power of 0 is 1, this gives us:
1 = x/(6x−1)
Now we can solve for x by cross-multiplying and moving all terms to one side to get a quadratic equation:
6x−x−1 = x
Which simplifies to:
5x−1 = 0
Adding 1 to both sides and then dividing by 5 gives us:
x = 1/5
The student should verify that the solution is in the domain of the original logarithmic functions.