SOLUTION: Solve the following equations: {{{ ln(x+4)=ln(x-8)-ln(x+1) }}} {{{ log6 (2-x) + log6 (11-x) = 2 }}}

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve the following equations: {{{ ln(x+4)=ln(x-8)-ln(x+1) }}} {{{ log6 (2-x) + log6 (11-x) = 2 }}}      Log On


   

Question 1001259: Solve the following equations:
+ln%28x%2B4%29=ln%28x-8%29-ln%28x%2B1%29+
+log6+%282-x%29+%2B+log6+%2811-x%29+=+2+
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

+ln%28x%2B4%29=ln%28x-8%29-ln%28x%2B1%29+
+ln%28x%2B4%29=ln%28%28x-8%29%2F%28x%2B1%29%29+........if log same, we have
+%28x%2B4%29=%28x-8%29%2F%28x%2B1%29+.......solve for x
+%28x%2B4%29%28x%2B1%29=%28x-8%29+
+x%5E2%2Bx%2B4x%2B4=x-8+
+x%5E2%2Bcross%28x%29%2B4x%2B4-cross%28x%29%2B8=0+
+x%5E2%2B4x%2B12=0+....use quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-4+%2B-+sqrt%28+4%5E2-4%2A1%2A12+%29%29%2F%282%2A1%29+
x+=+%28-4+%2B-+sqrt%28+16-48+%29%29%2F2+
x+=+%28-4+%2B-+sqrt%28+-32+%29%29%2F2+
x+=+%28-4+%2B-+sqrt%28+-16%2A2+%29%29%2F2+
x+=+%28-4+%2B-+4i%2Asqrt%28+2+%29%29%2F2+
x+=+%28-2+%2B-+2i%2Asqrt%28+2+%29%29+
solutions:
x+=+%28-2+%2B+2i%2Asqrt%28+2+%29%29+
or
x+=+%28-2+-+2i%2Asqrt%28+2+%29%29+



+log%286+%2C%282-x%29%29+%2B+log%286%2C+%2811-x%29%29+=+2+....change the base
+log%28%282-x%29%29%2Flog%28%286%29%29++%2B+log%28+%2811-x%29%29%2Flog%28%286%29%29+=+2+
+%28log%28%282-x%29%29++%2B+log%28+%2811-x%29%29%29%2Flog%28%286%29%29+=+2+
+%28log%28%282-x%29%29++%2B+log%28+%2811-x%29%29%29+=+2log%28%286%29%29+
+log%28%282-x%29%2811-x%29%29+=+log%28%286%5E2%29%29+
+log%28%282-x%29%2811-x%29%29+=+log%28%2836%29%29+.....if log same, we have
+%282-x%29%2811-x%29+=+36+....solve for x
+22-2x-11x%2Bx%5E2+=+36+
+x%5E2-13x+%2B22-36=0+
+x%5E2-13x+-14=0+
+x%5E2%2Bx-14x+-14=0+
%28+x%5E2%2Bx%29-%2814x+%2B14%29=0+
x%28+x%2B1%29-14%28x+%2B1%29=0+
%28x-14%29%28x+%2B1%29=0+
solutions:
if %28x-14%29=0+=>x=14-> disregard it it will make log of negative number which is undefined
if %28x+%2B1%29=0+=>highlight%28x=-1%29


Answer by ikleyn(52785) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve the following equations:
+ln%28x%2B4%29=ln%28x-8%29-ln%28x%2B1%29+
-----------------------------------

+ln%28x%2B4%29 = ln%28x-8%29 - ln%28x%2B1%29 ----->

x+%2B+4 = %28x-8%29%2F%28x%2B1%29       ----->

(x+4)*(x+1) = x-8,

x%5E2+%2B+5x+%2B+4 = x-8,

x%5E2+%2B+4x++%2B+12 = 0.

Discriminant  d = 4^2 - 4*1*12 = 16 - 48 = -44  is negative     ----->   no real solutions.

Answer.  There is no real solutions.