Solve for x:
(log(x + 5))/(log(3)) = 1 - (log(x + 3))/(log(3))
Subtract 1 - (log(x + 3))/(log(3)) from both sides:
-1 + (log(x + 3))/(log(3)) + (log(x + 5))/(log(3)) = 0
Bring -1 + (log(x + 3))/(log(3)) + (log(x + 5))/(log(3)) together using the common denominator log(3):
-(log(3) - log(x + 3) - log(x + 5))/(log(3)) = 0
Multiply both sides by -log(3):
log(3) - log(x + 3) - log(x + 5) = 0
log(3) - log(x + 3) - log(x + 5) = log(3) + log(1/(x + 3)) + log(1/(x + 5)) = log(3/((x + 3) (x + 5))):
log(3/((x + 3) (x + 5))) = 0
Cancel logarithms by taking exp of both sides:
3/((x + 3) (x + 5)) = 1
Take the reciprocal of both sides:
1/3 (x + 3) (x + 5) = 1
Multiply both sides by 3:
(x + 3) (x + 5) = 3
Expand out terms of the left hand side:
x^2 + 8 x + 15 = 3
Subtract 15 from both sides:
x^2 + 8 x = -12
Add 16 to both sides:
x^2 + 8 x + 16 = 4
Write the left hand side as a square:
(x + 4)^2 = 4
Take the square root of both sides:
x + 4 = 2 or x + 4 = -2
Subtract 4 from both sides:
x = -2 or x + 4 = -2
Subtract 4 from both sides:
x = -2 or x = -6
(log(x + 5))/(log(3)) ⇒ (log(5 - 6))/(log(3)) = (i π)/(log(3)) ≈ 2.8596 i
1 - (log(x + 3))/(log(3)) ⇒ 1 - (log(3 - 6))/(log(3)) = -(i π)/(log(3)) ≈ -2.8596 i:
So this solution is incorrect
(log(x + 5))/(log(3)) ⇒ (log(5 - 2))/(log(3)) = 1
1 - (log(x + 3))/(log(3)) ⇒ 1 - (log(3 - 2))/(log(3)) = 1:
So this solution is correct
The solution is:
Answer: |x = -2
+118677 log3(x+5)=1-log3(x+3)
Thanks guest, I just want to look too :)
\(log_3(x+5)=1-log_3(x+3)\\ log_3(x+5)+log_3(x+3)=1\\ log_3[(x+5)(x+3)]=1\\ log_3[(x+5)(x+3)]=log_33\\ (x+5)(x+3)=3\\ x^2+8x+15=3\\ x^2+8x+12=0\\ (x+6)(x+2)=0\\ x=-6 \qquad or \qquad x=-2\\ \text{But you cannot find the log of a negative number so -6 is no good}\\ x=-2\\ check\\ LHS=log_33=1\\ RHS=1-log_31=1\\good:)\)
x=-2 just as our first guest already determined. :)
