solve the equation. Check for extraneous solutions.

Question

ln x + ln(x+2)= 3

Answer 1

Hello!

Adding two logs of the same type (in this case logs of base e, 'natural logs') is the same as making them into one log and multiplying the insides.

So first combine the logs:

ln(x) + ln(x+2) =

ln[x*(x+2)] =

ln(x^2+2x) = 3

Next let's get rid of the log by converting it into an exponential equation. Remember the base (e) raised to the power (3) equals whatever is inside of the logarithm (x^2+2x)

So

e^3 = x^2 + 2x

e^3 is just a number (an infinite irrational number, but still a number), so we can simplify this by completing the square. Take the coefficient in front of the linear term (2 in front of x), cut it in half and square it. That's 1, so let's add 1 to both sides:

e^3 + 1 = x^2 + 2x + 1

This still makes the equation true and technically the same since we did the same thing to both sides. BUT now we can simplify the quadratic right side to be:

e^3 +1 = (x+1)^2

Finally, solve for x. Take the square root of both sides, then subtract 1, and that leaves one lone x by itself, and gives you the answer:

x = [sqrt(e^3+1)] - 1 or roughly 3.59

If that doesn't make sense please ask and I can record an explanation that you can hear!