Question

What is the true solution to the equation below? In e^(lnx) + ln(e^(lnx²)) = 2 ln 8

a) x = 2
b) x = 3
c) x = 4
d) x = 5

Answer 1

The true solution to the equation ln(x) + ln(e^(ln(x²))) = 2 ln(8a) is x = 4 (option c).

Step-by-step explanation:

To solve the given equation, let's simplify the expression step by step. We start with ln(x) + ln(e^(ln(x²))), using the property ln(a) + ln(b) = ln(ab), which simplifies to ln(xe^(ln(x²))). Next, we apply the property e^(ln(a)) = a, resulting in ln(x³).

Now the equation becomes ln(x³) = 2 ln(8a). Applying the property ln(a^b) = b ln(a), we simplify to 3 ln(x) = 2 ln(8a). Solving for x, we get x = e^(2/3 ln(8a)). Further simplifying, x = e^(ln((8a)^(2/3))), and using the property e^(ln(a)) = a, we arrive at x = (8a)^(2/3).

Finally, setting (8a)^(2/3) = 4 and solving for a, we find x = 4. Therefore, the correct solution is x = 4 (option c).