Question about changing a logarithm's base I've been using

Harold Hoover

Harold Hoover

Answered question

2022-03-30

Question about changing a logarithm's base
I've been using the following method to derive/remember the logarithm base conversion formula: If I want to convert loga(x) to an expression in base b, I say,
aloga(x)=x
logb(aloga(x))=logb(x)
loga(x)logb(a)=logb(x)
loga(x)=logb(x)logb(a)
It kind of feels like I'm working backwards, and I was wondering if there's a more direct way to go about it. I tried:
loga(x)=loga(blogb(x))
loga(x)=logb(x)loga(b)
but couldn't rid myself of the loga in the right-hand side of the equation. It occurred to me that if I could rewrite the "a" subscript as "blogb(a)", I might be onto something. (Or might not.) Does the notation ever get used like that, where you perform a substitution in a subscript?

Answer & Explanation

Korbin Rivera

Korbin Rivera

Beginner2022-03-31Added 11 answers

Deriving and remembering are different things. If all you want is to remember, do remember this: All logarithm functions are proportional to each other. Thus
loga(x)=Clogb(x)
for some constant C. To find out the value of C, insert x=b and remember that logb(b)=1
Dixie Reed

Dixie Reed

Beginner2022-04-01Added 15 answers

It is not common, but substitution is universal. Anywhere there is something that represents a number, you may put there anything else that represents the same number. This is at the very heart of mathematics, the fact that numbers transcend any representation there of. Oh, and I have no idea how that would help, unless you know how to extract powers from a base. (If you do, show me your powers of magic in the comments. I have been interested in logarithms recently.)

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