Final answer:
The correct solution to the given logarithmic equation is found by using the property of logarithms relating to the division of two numbers and simplifying the expression to isolate and solve for x. The final answer is C. x=4/9.
Step-by-step explanation:
Solving the Logarithmic Equation
To find the true solution to the logarithmic equation log₂(6x) - log₂(√x) = 2, we can use the properties of logarithms. Specifically, we use the property that states the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator (log(a/b) = log a - log b). We can simplify the original equation by using this property:
• Combine the logarithms: log₂( 6x/√x).
• Simplify the expression under the logarithm: log₂(6x/x^(1/2)).
• Since x^(1/2) is the square root of x, the expression simplifies to log₂(6√x).
• Setting up the equation: log₂(6√x) = 2.
• Convert the logarithmic equation into an exponential equation: 6√x = 2².
• Solve for x: 6√x = 4.
• Isolate √x: √x = 4/6.
• Simplify the fraction: √x = 2/3.
• Square both sides to solve for x: x = (2/3)².
• Calculate the value of x: x = 4/9.
Therefore, the correct answer is C. x=4/9, which is our true solution.