Final answer:
To solve ln(x-4) + ln(4x+1) = 9, we combine the logarithms, exponentiate, and then use the quadratic formula to find the value of x, ensuring we consider the domain restrictions of the original logarithmic equations.
Step-by-step explanation:
To find x for the equation ln(x-4) + ln(4x+1) = 9, we first use the property of logarithms that allows us to combine two logs with the same base, which are being added, into a single log by multiplying their arguments:
ln((x-4)(4x+1)) = 9.
Next, we exponentiate both sides of the equation to cancel out the natural logarithm, obtaining the equation (x-4)(4x+1) = e^9.
Expanding the left side gives us a quadratic equation: 4x^2 + x - 4 = e^9. This quadratic equation can be solved by applying the quadratic formula, x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = 4, b = 1, and c = -e^9.
After calculating the values of b^2 - 4ac and sqrt(b^2 - 4ac), we substitute them into the quadratic formula to find the two potential values for x. However, we must also consider the domain of the original logarithmic expressions, which implies that x must be greater than 4, thus we can discard any solution less than 4 or any non-real solutions.