What does log_a(b) equal to? Does log_a(b)=(log_c(b))/(log_c (a)) or log_a(b)=(ln (b))/(ln (a)) Is there any difference between the two?

Alice Chen

Alice Chen

Answered question

2022-11-14

What does log a ( b ) equal to?
Does
log a ( b ) = log c ( b ) log c ( a )
or
log a ( b ) = ln ( b ) ln ( a )
??
Is there any difference between the two?

Answer & Explanation

Calvin Maddox

Calvin Maddox

Beginner2022-11-15Added 15 answers

Suppose
log a ( b ) = x .
Then, by definition, we have
b = a x ,
where a > 0
Now let c > 0
Taking the logarithm to the base c of both sides of the equation b = a x , we get
log c ( b ) = log c ( a x ) .
Or
log c ( b ) = x log c ( a )
using the property of the logarithm.
So, if log c ( a ) 0, then upon dividing both sides of the last equation by log c ( a ), we get
x = log c ( b ) log c ( a ) .
Or
log a ( b ) = log c ( b ) log c ( a ) .
Now taking c to be equal to e in the last relation, we get
log a ( b ) = ln ( b ) ln ( a ) .
Aron Heath

Aron Heath

Beginner2022-11-16Added 3 answers

ln ( a ) is just a shorter way to write log e ( a ) so the second formula is an instance of the first with c = e, both identities are right

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