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how do you solve ln(x+1)-ln(x)=ln4

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2 Answers

ln(x+1) -ln(x) = ln(4)
 
if you remember the basic rules of logs
 
1) logb(mn) = logb(m) + logb(n)

2) logb(m/n) = logb(m) – logb(n)

3) logb(mn) = n · logb(m)
 
So if we look at your eqn, we can use rule 2 
 
ln(x+1) - ln(x) = ln((x+1)/x)) = ln 4
 
Then if you remember that to solve an eqn for a given variable you need to undo whatever has been done to the variable. In our case, utilizing the inverse of the ln(x)
 
f (f -1(x)) = eln(x) = x
 
we can apply that to both sides of the eqn
 
eln((x+1)/x)) = eln 4  ⇒  (x+1)/x = 4
 
then solving this for x is easier
 
(x+1)/x) = 4
(x+1)/x •  x = 4 • x
x+1 = 4x
x+1 -x = 4x -x
1 = 3x
1/3 = 3x/3
x = 1/3
 
Hope this helps
ln[(x+1)/x] = ln4
So, (x+1)/x = 4
x+1 = 4x
Answer: x = 1/3