Solve the logarithmic equation lnx - ln(x+1) = 2?
- print Print
- list Cite
Expert Answers
calendarEducator since 2008
write3,662 answers
starTop subjects are Math, Science, and Social Sciences
ln x -ln (x+1) = 2
since ln x-ln y = ln x/y
then ln(x/(x+1)= 2
==> e^2 = x/(x+1)
==> (x+1)e^2 = X
==> xe^2 + e^2 = x
==> xe^2 -x= -e^2
==> x(e^2-1) = - e^2
==> x = -e^2/(e^2 -1)
Related Questions
- Solutions for logarithmic equation lnx+ln(x+1)=ln2
- 1 Educator Answer
- Simplify: i) ln(2x) + ln(2/x) ii) ln(x^2 - 1)- ln(x-1) Then solve: i) lnx = 5 ii) lnx + ln5 =...
- 2 Educator Answers
- solve for x : lnx + ln(3x-2)=0
- 2 Educator Answers
- Solve the equation : 2*sin^2 x + cos x - 1 = 0
- 1 Educator Answer
- solve the equation: 2^(2x-1)= 8^x
- 1 Educator Answer
ln is used for denoting natural logarithm that is logarithm to the base e.
To solve the equation ln x - ln (x+1) = 2, use the property of logarithm, log a - log b = log(a/b)
This gives:
`ln (x/(x +1)) = 2`
`log_e (x/(x +1)) = 2`
Now log_b a = c gives a = b^c
The equation can be written as:
`x/(x +1) = e^2`
`(x+1)/x = 1/e^2`
`1 + 1/x = 1/e^2`
`1/x = (1/e^2 - 1)`
`x = 1/(1/e^2 - 1)`
The solution of the equation `ln x - ln (x+1) = 2` is `x = 1/(1/e^2 - 1)`
To solve: lnx-ln(x+2) = 2.
Solution:
LHS = ln {x/(x+2)} = 2. Taking anti logarithms on both sides,
x/(x+2) = e^2. Or
x = e^2(x+2) . Or
x-xe^2 = 2e^2. Or
x(1-e^2) = 2e^2. Or
x = 2e^2/(1-e^2)
For the beginning, we'll use the quotient property of the logarithms:
lnx - ln(x+1) = ln [x/(x+1)]
Now, we'll have to use the one to one property, that means that:
ln [x/(x+1)] = 2 lne if and only if [x/(x+1)] = e^2
After cross multiplying, we'll get:
x = x*e^2 + e^2
We'll move the terms which are containing the unknown, to the left side:
x - x*e^2 = e^2
After factorizing, we'll get:
x (1-e^2) = e^2
x = e^2/(1-e^2)
But 1-e^2<0, so x = e^2/(1-e^2)<0, which is impossible because x has to be positive!
So, the equation has no solutions.
Student Answers