Why can't you add a coefficient before the logarithm base change rule? Why is it that the rule log_b(x)= (log_c(x))/(log_c(b)) (the logarithm base change rule) is true but a log_b(x)=(a log_c(x))/(a log_c(b))

Valery Cook

Valery Cook

Answered question

2022-10-28

Why can't you add a coefficient before the logarithm base change rule?
Why is it that the rule
log b ( x )   =   log c ( x ) log c ( b )  
(the logarithm base change rule) is true but
  a log b ( x )   =   a log c ( x ) a log c ( b )  
isn't?
For example why does the equation,
log 49 3   =   log 49 log 3  
work but
  4 log 49 3   =   4 log 49 4 log 3
does not?
All you are doing is multiplying the logarithm by a term, and still multiplying each part by it when "splitting the logarithm up."
P.S. If the answer could be explained in somewhat simple terms it would be appreciated. I'm not like a math major or anything. :)

Answer & Explanation

exalantaswo

exalantaswo

Beginner2022-10-29Added 14 answers

You have
log b ( x ) = log c ( x ) log c ( b ) .
Multiplying on both sides by a gives
a log b ( x ) = a ( log c ( x ) log c ( b ) ) = a 1 ( log c ( x ) log c ( b ) ) = a log c ( x ) log c ( b ) .
Your mistake was to multiply on the left by a but on the right by a a , which is equal to 1.

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