Why does the log(12.5) and the log(0.8) have the same repeating decimal sequence?

Question

Referring to log base 10

Answer 1

log(12.5) is positive
log(0.8) is negative

If we add them we would expect an integer result if they had the same decimal.

Remember this rule of logs:
log(a) + log(b) = log(ab)

log(12.5) + log(0.8)
log(12.5 * 0.8)
log(10)
= 1
This is an integer result.

Because:
log(12.5) + log(0.8) = 1

The answer is yes, they have the same repeating decimal sequence.

Answer 2

using the log rules and the fact that:
10/0.8 = 12.5

we get:

log 12.5 = log 10/0.8 = log 10 - 0.8 = 1 - log 0.8

Answer 3

I'm assuming that you are taking logs in base 10. Use the fact that log(a/b) = log(a) - log(b)

log(12.5) = log(100/8) = log(100) - log(8) = 2 - log(8)
log(0.8) = log(8/10) = log(8) - log(10) = log(8) - 1

Note that both terms are a combination of an integer and
log(8)

Answer 4

They have the same decimal sequence because their mantissas are identical in base 10.
Log(12.5) = Log(10) + log(1.25) = 1 + log(1.25)
0.8 is the reciprocal of 1.25, so
Log(0.8) = - log(1.25)
Adding the 1 to log(1.25) does not change its decimal part.

Answer 5

log 12.5 = log (100/8) = 2 - log8
log 0.8 = log(8/10) = log8 - 1

I don't know is there any relate to ur question

Answer 6

Puzzling was right, except that the numbers a non-repeating irrationals. They go on forever without repetition.