Is there a logarithm function through this three given points? I've got the task to find a logarith

il2k3s2u7

il2k3s2u7

Answered question

2022-05-26

Is there a logarithm function through this three given points?
I've got the task to find a logarithm function which contains the following points:
A ( 5 4 ) B ( 3 6 ) C ( 2 8.5 )
Now I need to find the logarithm function. My idea was to use f ( x ) = a log ( x b ) + c and a system of equations:
4 = a log ( 5 b ) + c 6 = a log ( 3 b ) + c 8.5 = a log ( 2 b ) + c
I tried to solve for a, b and c without success. Does someone have a hint for me?

Answer & Explanation

Conor Frederick

Conor Frederick

Beginner2022-05-27Added 7 answers

Found a way easier and more effective way (facepalm)
By subtracting equations, we find
2 = a log ( 3 b 5 b ) 4.5 = a log ( 2 b 5 b ) 2.25 = log ( 2 b 5 b ) log ( 3 b 5 b ) ( 3 b 5 b ) 9 / 4 = 2 b 5 b ( 3 b ) 9 / 4 = ( 2 b ) ( 5 b ) 5 / 4 ( 3 b ) 9 = ( 2 b ) 4 ( 5 b ) 5
This makes b the root of an 8 t h degree polynomial:
0 = 30317 90951 b + 116268 b 2 82764 b 3 + 35907 b 4 9735 b 5 + 1614 b 6 150 b 7 + 6 b 8
Now that you have one variable ( b 1.44969), it should be simple to solve the rest (that's not to say there will be nice ways to represent a and c)
Kendrick Pierce

Kendrick Pierce

Beginner2022-05-28Added 3 answers

4 c a = log ( 5 b ) 6 c a = log ( 3 b ) 8.5 c a = log ( 2 b )
So we obtain
b = 5 e 4 c a = 3 e 6 c a = 2 e 8.5 c a 2 = e 4 c a e 6 c a = e 4 c a ( 1 e 2 a ) log 2 = 4 c a + log ( 1 e 2 a ) c = 4 + a ( log ( 1 e 2 a ) log 2 )
Similarly, you may obtain c = 4 + a ( log ( 1 e 4.5 a ) log 3 )
So we have
log ( 1 e 2 a ) log 2 = log ( 1 e 4.5 a ) log 3 1 2 ( 1 e 2 a ) = 1 3 ( 1 e 4.5 a ) 1 6 = 1 2 e 2 a 1 3 e 4.5 a
A Computer Algebra System will tell you a 2.41377. From there you can deduce b and c

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