How to move from powers to simple logarithms? I'm following a book that briefly moves from 16

Dale Tate

Dale Tate

Answered question

2022-06-13

How to move from powers to simple logarithms?
I'm following a book that briefly moves from
16000 × 2 ( x 24 ) = 1600
to
x = 24 ( log ( 2 ) + log ( 5 ) ) log ( 2 )
adding the comments that
log ( 1600 ) = 6 log ( 2 ) + 2 log ( 5 ) log ( 16000 ) = 7 log ( 2 ) + 3 log ( 5 )
What general principles are used to achieve this and how to spot possible ways to apply a similar operation in the future?

Answer & Explanation

jarakapak7

jarakapak7

Beginner2022-06-14Added 14 answers

In general:
log a b = b log a log ( a b ) = log a + log b
Specific to the above problem:
log ( 1600 ) = log ( 2 6 5 2 ) = log ( 2 6 ) + log ( 5 2 ) = 6 log 2 + 2 log 5
log ( 16000 ) = log ( 2 7 5 3 ) = 7 log 2 + 3 log 5
So,
16000 × 2 x / 24 = 1600 log 16000 + ( x / 24 ) log 2 = log 1600 ( x / 24 ) log 2 = log 2 log 5 x = 24 log 2 + log 5 log 2

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