solving exponential equations: 2 lnx - ln(x+1) = 0?

Question

Please provide/show all work for this problem

Help me with this one: 2 lnx - ln(x+1) = 0

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I know that you should exponetiate the 2 in the following example
2 log x - log(x+2)=0
--logx^2 - log(x+2)^2=0
-----x^2=x+2
--------x=2

Answer 1

2lnx - ln(x+1) = 0
lnx^2 - ln(x+1) = 0
ln(x^2/(x+1) = 0
e^(x^2/(x+1)) = e^0
x^2/(x+1) = 1
x^2 = 1*(x+1) assuming x<>-1
x^2 - x - 1 = 0
using quadratic formula,
x=(1 +/- (5)^.5)/2

(edited to fix my misuse of the quadratic formula)

Answer 2

To solve these kind of problems you use the logarithm properties for sums, substraction, power, etc. Look in the book...

For example:

2 lnx - ln(x+1) = 0
can be transformed into:

ln (x^2/(x+1) = 0
using such properties and as ln A = 0 ===> A = 1

Then
x^2 / (x+1) = 1 and
x^2 = x + 1
and you are left to solve the quadratic equation
x^2 - x - 1 = 0

Answer 3

Actually you dont know that.

Thats the point you are missing. In logarithms.. exponentiation is turned into multiplication.

log(a^b) = b* log(a)

so lets generalize your question, because it looks like you might be looking at the mechanics, and not the idea behind them.

a*log(x) - log(x+b) = 0
is
log(x^a) - log(x+b) = 0
therefore
log(x^a) = log(x+b)

exponentiating both sides of the equation
exp(log(x^a)) = exp( log(x+b))
is
x^a=x+b

or
x^a -x - b =0

Key ideas here:
- properties of logs allow you to exchange:
- > exponentiation and multiplication
- > multiplication and addition

- whatever you do to one side you do to the other
- teachers give new problems that often reduce to older/familiar problems, like the quadratic in your example.
- a zero on one side of an equation is meant to be used.

Answer 4

2 lnx - ln(x+1) = 0
or, logx^2 - log(x+2)=0
or log(x^2 / (x+2))=0 = log1
or (x^2 / (x+2)) = 1
or x^2 = (x+2)

x^2 - x - 2 = 0
or x = 0.5 ( 1 ± 3)
or x = 2, -1

Answer 5

you can basically do the same thing you did with the log:
lnx^2-ln(x+1)=0
lnx^2=ln(x+1)
x^2=x+1
x^2-x-1=0
solve using quad formula and you get
x = +/- (sqrt(5)+1)/2

Hope that helps!