How do you solve a logarithm with a non-integer base?How would one calculate the log...

Raul Walker

Raul Walker

Answered

2022-07-14

How do you solve a logarithm with a non-integer base?
How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example:
0.5 x = 8  (where  x = 3 )
log 0.5 8 = 3
How would you solve this, and how would this work for an irrational base (like 2 )?

Answer & Explanation

vrtuljakc6

vrtuljakc6

Expert

2022-07-15Added 16 answers

Let's rewrite this in a different way:
0.5 x = 8
Take the logarithm with respect to any base a > 0
log a ( 0.5 x ) = log a 8
which becomes
x log a 0.5 = log a 8
or
x = log a 8 log a 0.5
You would stop here weren't from the fact that 8 = 2 3 and 0.5 = 2 1 so.
x = log a 8 log a 0.5 = log a 2 3 log a 2 1 = 3 log a 2 log a 2 = 3
You need to compute no logarithm, actually.
invioor

invioor

Expert

2022-07-16Added 3 answers

This is the base changing formula :
log a ( x ) = log b ( x ) log b ( a )

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