lnx + ln(x+3) = 2
ln(x(x+3)) = 2
x(x+3) = e^2
x^2 + 3x - e^2 = 0
solve using the quadratic formula
a=1
b=3
c=e^2
make sure to check your answers, since x > 0
To solve the equation lnx + ln(x+3) = 2, you can use logarithmic properties to simplify it first.
1. Combine the logarithms using the multiplication property of logarithms: ln(x) + ln(x+3) = ln(x(x+3)) = 2
2. Rewrite the equation in exponential form: e^2 = x(x+3)
3. Simplify the exponential equation to a quadratic equation: e^2 = x^2 + 3x
4. Rearrange the equation to form a quadratic equation set equal to zero: x^2 + 3x - e^2 = 0
Now, to solve the quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation x^2 + 3x - e^2 = 0, the coefficients are:
a = 1, b = 3, and c = -e^2
Substituting these values into the quadratic formula, we get:
x = (-(3) ± √((3)^2 - 4(1)(-e^2))) / (2(1))
Simplifying further:
x = (-3 ± √(9 + 4e^2)) / 2
Please note that 'e' represents Euler's number, approximately equal to 2.71828.
By evaluating the expression (-3 ± √(9 + 4e^2)) / 2, you will obtain the solutions to the equation lnx + ln(x+3) = 2.