Solve ln(4x+3) - ln(x²-8) = 2: Finding the Value of x

Let's solve the logarithmic equation step-by-step:

Step 1: Combine the Logarithms
Using the property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$, we combine the logarithms:

$\ln\left(\frac{4x+3}{x^2-8}\right) = 2$

Step 2: Remove the Logarithm by Exponentiation
Exponentiate both sides with base $e$ to get rid of the natural logarithm:

$\frac{4x+3}{x^2-8} = e^2$

Step 3: Solve the Resulting Equation
Multiplying both sides by $x^2 - 8$ to eliminate the fraction:

$4x + 3 = e^2(x^2 - 8)$

Expanding and rearranging gives us:

$e^2x^2 - 4x - 8e^2 - 3 = 0$

Let's employ the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = e^2$, $b = -4$, and $c = -(8e^2 + 3)$.

Calculate the discriminant:

$b^2 - 4ac = (-4)^2 - 4(e^2)(-(8e^2 + 3))$

Solving this using numerical approximations (since we have $e^2 \approx 7.39$), you get:

$x \approx 3.18$

Conclusion:
The value of $x$ is approximately $3.18$, which confirms our choice.