Hi xkyroutx-ga,
Natural logarithm (ln) can not be distributed like what you have done:
you can not separate ln(x+1) into ln(x)+ln(1).
The following are some logarithmic identities that you should know:
1. log(A*B) = log(A)+log(B)
2. log(A/B) = log(A)-log(B)
3. log(A^B) = B * log(A)
Now, let's look at your equation:
ln(x+1)-ln(x) = e
By identity #2 above, you can simplify ln(x+1)-ln(x) into ln((x+1)/x), giving:
ln((x+1)/x) = e
The next is to take the exponent of e on both sides:
e^(ln((x+1/x) = e^e
(x+1)/x = e^e, x not equal to 0
now we multiply both sides by x, since x is not 0:
x+1 = x * e^e
and subtract x from both sides:
1 = x * e^e - x
1 = x(e^e - 1)
and divide both sides by e^e - 1:
x = 1/(e^e - 1)
Now to check the answer:
x + 1 = 1/(e^e-1) + 1
x = 1/(e^e-1)
(x+1)/x = (1/(e^e-1) + 1)*(e^e-1)
(x+1)/x = 1 + e^e - 1 = e^e
ln ((x+1)/x) = ln (e^e) = e
So your answer is x = 1/(e^e - 1).
Reference:
Logarithmic identities:
http://en.wikipedia.org/wiki/Logarithm#Easier_computations
I hope that this was clear. If you need clarification, please use the
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secret901-ga |