Final answer:
To solve the equation ln(x) + ln(4x) = 2, we can combine the logarithms to get ln(4x^2) = 2. Rewriting in exponential form gives e^2 = 4x^2. Solving for x, we find x = e/2. However, when we substitute this solution back into the original equation, it does not hold. Therefore, there are no real solutions to the equation.
Step-by-step explanation:
To solve the equation ln(x) + ln(4x) = 2, we can use the properties of logarithms to combine the two logarithms into one. The sum of two logarithms with the same base is equal to the logarithm of their product. So, we have ln(x * 4x) = 2. Simplifying, we get ln(4x^2) = 2.
To solve for x, we can rewrite the equation in exponential form. The natural logarithm function and the exponential function are inverse functions, so ln(4x^2) = 2 can be written as e^2 = 4x^2. Taking the square root of both sides, we get e = 2x. Solving for x, we get x = e/2.
Now, we need to check for extraneous solutions. Exponential equations can sometimes produce solutions that don't satisfy the original equation. In this case, substituting x = e/2 into the original equation gives:
ln(e/2) + ln(4(e/2)) = 2
Simplifying, we get ln(e/2) + ln(2e) = 2. Using properties of logarithms, we can combine the two logarithms: ln(2e^2) = 2. However, this equation is not true for any value of e, so the solution x = e/2 is extraneous and there are no real solutions to the equation.