Solution of an exponential equation Probably very simple question. Why the solution of 1=n(1−a)^t in terms of t is equal to: t=(ln n)/(ln 1/(1-a))

Siemensueqw

Siemensueqw

Answered question

2022-11-17

Solution of an exponential equation
Probably very simple question. Why the solution of
1 = n ( 1 a ) t
in terms of t is equal to:
t = ln n ln 1 1 a

Answer & Explanation

Hector Hamilton

Hector Hamilton

Beginner2022-11-18Added 13 answers

we can write 1 n = ( 1 a ) t taking the logarithm of both sides we get
ln ( n ) = t ln ( 1 a )
and thus we get
t = ln ( n ) ln ( 1 a )
Kareem Mejia

Kareem Mejia

Beginner2022-11-19Added 9 answers

ln ( 1 ) = 0 = ln ( n ( 1 a ) t )
Now the later can be expanded using
ln ( a b ) = ln a + ln b
Thus
(*) ln ( n ( 1 a ) t ) = ln n + ln [ ( 1 a ) t ]
Using
ln a b = b ln a
We can rewrite Eq * as
ln n + t ln ( 1 a ) = 0
You can rearrange for t now. Now the final bit of the puzzle is using the fact
ln ( 1 a ) = ln ( 1 1 a )

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