Logarithmic problems help?

Question

Im stuck on a few problems, please help me

solve for x
e^(x+4)=1/[e^(2x)]
e^(4x)-3e^(2x)-18=0
ln(x+2)-lnx=ln(x-5)-ln(x-1)

Answer 1

Problem 1
e^(x+4)=1/[e^(2x)] multiply both sides by e^(2x)
becomes [e^(x+4)][e^(2x)] = 1 using (e^a)(e^b) = e^(a+b)
e^[(x+4)+(2x)] = 1 Take the natural log of both sides
LN(e^[(x+4)+(2x)]) = Ln(1) we know that Ln(1) = 0
(x+4)+(2x) = 0
3x+4 = 0
Solving for x... x = -4/3

Problem 2
e^(4x)-3e^(2x)-18=0 Using x^(ab) = (x^a)^b or (x^b)^a
(e^(2x))^2-3e^(2x)-18=0 similar to a quadratic in terms of (e^(2x))
[(e^(2x))-6][(e^(2x))+3] = 0
solving for (e^(2x))
(e^(2x)) = 6 and (e^(2x)) = -3
Now solving for x by taking Ln of each solution
Ln(e^(2x)) = Ln(6)
2x = Ln(6)
x = Ln(6)/2
and
Ln(e^(2x)) = Ln(-3)... this answer does not exist since no such thing as Ln(negative number) so Ln(-3) is undefined

Problem 3
ln(x+2)-lnx=ln(x-5)-ln(x-1) we know that ln(a) - ln(b) = ln(a/b)
rewriting the equation we get
Ln((x+2)/(x)) = Ln((x-5)/(x-1)) take the exponent of both sides e(ln(a)) = a
(x+2)/(x) = (x-5)/(x-1) cross multiply both sides by the x(x-1)
(x+2)(x-1) = (x-5)(x) factor
x^2 + x -2 = x^2 -5x "move" the values to the left
x^2 + x -2 - x^2 +5x = 0 simply
6x -2 = 0 solve for x
x = 1/3

Answer 2

e^(x+4)= e^-2x

take log of both sides

(x+4) = -2x ==> x = -2

NEXT:

e^(4x)-3e^(2x)-18=0

goes to

e^(2x)^2 - 3e^(2x) - 18 = 0

which looks like

u^2 - 3u - 18 = 0

when u = e^2x

now use the quadratic formula

e^2x =( 3 +- sqrt(9 - 4*(-18))/ 2

e^2x = (3+- 9) /2 = 12/2 or -6/2

the negative answer will not make sense

e^2x = 6

2x = ln6

x= ln/2

NEXT:

ln((x+2)/x) = ln((x-5)/(x-1)) From the law ln (a/b) = ln(a) - ln (b)

so take everything to the power "e"

(x+2)/x = (x-5)/ (x-1)

cross multiply

x^2 +x -2 = x^2 -5x

-4x = -2

x = 1/2

Answer 3

e^(x + 4) = 1/[e^(2x)]
e^(x + 4) = e^-(2x)
x + 4 = -2x
3x = -4
x = -4/3

e^(4x) - 3e^(2x) - 18 = 0
(e^(2x) - 6)(e^(2x) + 3) = 0
e^2x = -3, 6
2x = ln(6)
x = 0.896

ln(x + 2) - lnx = ln(x - 5) - ln(x -1)
(x + 2)/x = (x - 5)/(x - 1)
x^2 + x - 2 = x^2 - 5x
6x = 2
x = 1/3