Question 719670
The key to this problem is to understand the logarithmic properties. The property invoked in this problem is the multiplication property. That is ln(a) + ln(b) = ln(ab).

Using this we know that ln(x-3)+ln(3x+1) is really  ln[(x-3)(3x+1)].

So ln[(x-3)(3x+1)] = 10.

By FOIL:

ln[3x^2 -8x -3] = 10

Since ln is the natural log [with base e], we can EXPONENTIATE both sides by a base of e.

e^ln[3x^2-8x-3] = e^10

3x^2 - 8x - 3 = e^10

3x^2 - 8x - 3-e^10 = 0


Use the quadratic formula to clean this up:

{{{(8 +- sqrt(64 - (4*3*(-3-e^10))))/ 6)}}}

Which is roughly: 

x= 87.036, -84.369

Remember that domain of the natural log is (0, infinity). So -84.369 will give us a negative in the logarithm which is not allowed. So the only acceptable solution is 87.036 or in exact form:  {{{(8 + sqrt(100 + 12e^10))))/ 6)}}} which may be simplified to:

{{{(8 + sqrt(4*(25+3e^10)))/6}}}
{{{(8 + 2*sqrt(25+3e^10))/6}}}
{{{(4 + sqrt(25+3e^10))/3}}} <--- in simplest form