How do you solve #ln (x) + ln (x-2) = ln (3x+14)#?
1) Establish the domain
2) Simplify until you have a polynomial (often linear or quadratic) equation
3) Solve the quadratic equation
4) Determine the solutions w. r. t. the domain
1) Establishing the domain
First, let's find out the domain for which the logarithmic terms are defined.
# x > 0# # x - 2 > 0 => x > 2# #3x + 14 > 0 => x > - 14/3 #
The most restrictive one is
So, any possible solutions need to satisfy
Now, let's simplify your equation using the logarithmic rule
In your case, it means:
Now we can use that
3) Solving the quadratic equation
Here, the solutions are
Let me show you a different method though that works here too. It is especially easy if
The trick is to factorize your
and if you succeed doing so,
So, the goal is finding two integers
both hold at the same time.
It's easy to see that both equations
so you can factorize your equation as follows:
4) Determining the solution w.r.t. domain
Now, as we have stated that our domain is
Hope that this helped!