Question 446792
{{{1n (x+7) + 1n (x+3) = 1n 77}}}
First must definition domain:
X+7>0  and x+3>0 there for domain is x>-3
{{{1n (x+7) + 1n (x+3 ) = ln (x^2+10x+21)= ln 77}}}
x^2+10x+21=77
x^2+10x-56=0  
{{{x = (-10 +- sqrt( 10^2-4*1*(-56) ))/(2*1) }}}
{{{x1 = (-10 - sqrt( 324 ))/ 2 }}}
{{{x1= (-10-18)/2=-14}}} that is not acceptable and
{{{x2 = (-10 + sqrt(324))/ 2 }}} 
{{{x2=(-10+18)/2=4}}} is acceptable and is answer.
Other solution is 
ln a ln b= ln 77 find a and b that multiple are 77.
a=7 and b= 11 (or a=11 and b=7)! There for x=4