In the logarithmic equation ln(x^2 + 5x + 9) = ln 3, what value of x satisfies this equation?

To solve the logarithmic equation ( \ln(x^2 + 5x + 9) = \ln 3 ), the property of logarithms that states if ( \ln a = \ln b ), then ( a = b ) can be applied. By using this principle, we can rewrite the equation in its exponential form:

[

x^2 + 5x + 9 = 3

]

Next, we simplify the equation by transferring 3 to the left side:

[

x^2 + 5x + 6 = 0

]

This is a quadratic equation, which can be factored or solved using the quadratic formula. Factoring:

[

(x + 2)(x + 3) = 0

]

Setting each factor to zero reveals the solutions:

[

x + 2 = 0 \implies x = -2

]

[

x + 3 = 0 \implies x = -3

]

The solutions to the quadratic equation are ( x = -2 ) and ( x = -3 ).

However, it's critical to verify the solutions with regard to the domain of the logarithm function.