Solved on Nov 18, 2023

Find the value of xx that satisfies the equation ln(x3)+ln(3x+1)=8\ln(x-3) + \ln(3x+1) = 8.

STEP 1

Assumptions1. The equation is ln(x3)+ln(3x+1)=8\ln (x-3)+\ln (3x+1)=8 . We are solving for xx
3. The logarithm is a natural logarithm, meaning it is base ee

STEP 2

We can use the property of logarithms that states the sum of two logarithms is equal to the logarithm of their product. So, we can combine the two logarithms on the left side of the equation into one.
ln(x)+ln(x+1)=ln((x)(x+1))\ln (x-)+\ln (x+1)=\ln((x-)(x+1))

STEP 3

Now, we can rewrite the equation as followsln((x3)(3x+1))=8\ln((x-3)(3x+1))=8

STEP 4

To remove the logarithm, we can use the property of logarithms that states a=lnba=\ln b is equivalent to ea=be^a=b. In this case, we can rewrite the equation as follows(x3)(3x+1)=e8(x-3)(3x+1)=e^8

STEP 5

Now, we can simplify the equation by expanding the product on the left side of the equation.
3x28x3=e83x^2 -8x -3 = e^8

STEP 6

We now have a quadratic equation. To solve for xx, we can use the quadratic formula. But first, we need to move e8e^8 to the left side of the equation to set the equation equal to zero.
3x28x3e8=03x^2 -8x -3 - e^8 =0

STEP 7

Now, we can use the quadratic formula to solve for xx. The quadratic formula isx=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}where aa, bb, and cc are the coefficients of the quadratic equation.

STEP 8

In our equation, a=3a =3, b=8b = -8, and c=3e8c = -3 - e^8. Plugging these values into the quadratic formula gives usx=(8)±(8)243(3e8)23x = \frac{-(-8) \pm \sqrt{(-8)^2 -4*3*(-3 - e^8)}}{2*3}

STEP 9

olving the above equation will give us the values of xx. However, we need to remember that the domain of the original equation (due to the logarithms) is x>3x >3 and x>/3x > -/3, so we need to check if the solutions satisfy these conditions.

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord