Value of log_43(43!) ?
Answers
Answer:
43 is a prime number, so
log
43
cannot be expressed in terms of logarithms of smaller numbers.
Using a calculator, we find:
log
43
≈
1.63346845558
How could we find it by hand?
One somewhat arduous method goes as follow:
Note that
10
1
≤
43
<
10
2
, so the portion of the logarithm before the decimal point must be
1
Dividing
43
by
10
(and thus subtracting
1
) from the logarithm, we get
4.3
.
If we raise this to the tenth power, then its logarithm will be multiplied by
10
, i.e. shifted one place to the left.
We find:
4.3
10
=
2161148.2313284249
So:
10
6
≤
4.3
10
<
10
7
So the next digit of the logarithm is
6
Divide
2161148.2313284249
by
10
6
to subtract
6
from its logarithm and raise to the tenth power to shift the logarithm another place to the left:
2.1611482313284249
10
≈
2222.519
Then:
10
3
≤
2222.519
<
10
4
so the next digit is
3
Keep on going for as many digits as you want.
Thus far we have found:
log
43
≈
1.63
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