problem solving logarithmic equation and reaching an equivalence ok so i've had a problem trying to simplify the ln[sqrt(1+(u^2)/(a^2))+(u)/(a)] and this is supposed to be equal to : ln [ sqrt(a^2+u^2) + u ]

Aleah Booth

Aleah Booth

Answered question

2022-07-19

problem solving logarithmic equation and reaching an equivalence
ok so i've had a problem trying to simplify the ln [ 1 + u 2 a 2 + u a ] and this is supposed to be equal to : ln [ a 2 + u 2 + u ]
how is this posible ?? i've tried to solve this for more than 2 hours and couldn't get to this equivalence. any suggestions ?

Answer & Explanation

yermarvg

yermarvg

Beginner2022-07-20Added 19 answers

I expect that you got this as the result of an (indefinite) integration, and a is a constant. Let a be positive. We are taking the ln of
1 a ( u 2 + a 2 + u ) .
Taking the ln, we get
ln ( a 2 + u 2 + u ) ln a .
But ln a is a constant, so can be absorbed into the constant of integration.
In more detail, if
ln ( 1 + u 2 a 2 + u a ) + C
is the answer to an indefinite integral problem, where C is an arbitrary constant, then
ln ( a 2 + u 2 + u ) + D
is a correct answer to the same problem.
This sort of thing happens a lot, particularly with trigonometric functions. As a simple example, if sin 2 x + C is "the" answer to an indefinite integration problem, then so is cos 2 x + C
Aleah Booth

Aleah Booth

Beginner2022-07-21Added 5 answers

ln x is injective so, if ln x = ln y x = y. So, the following should hold true:
1 + u 2 a 2 + u a = a 2 + u 2 + u
However, simply plugging in u = 1 , a = 2 gives us
1 + 1 4 + 1 2 = 4 + 1 + 1 5 4 + 1 2 = 5 + 1 5 4 = 5 1 2 5 2 = 2 5 1 2 5 = 2 5 1
which is false.
Hence they are not the same.

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