How do you calculate log 1300log1300?
1 Answer
Explanation:
Suppose you know:
log 2 ~~ 0.30103log2≈0.30103
ln 10 ~~ 2.302585ln10≈2.302585
Then:
1300 = 10*2*65 = 10*2*64*65/64 = 10*2^7*(1+1/64)1300=10⋅2⋅65=10⋅2⋅64⋅6564=10⋅27⋅(1+164)
So:
log 1300 = log (10*2^7*(1+1/64))log1300=log(10⋅27⋅(1+164))
color(white)(log 1300) = log 10 + 7 log 2 + log(1+1/64)log1300=log10+7log2+log(1+164)
color(white)(log 1300) = log 10 + 7 log 2 + ln(1+1/64)/ln 10log1300=log10+7log2+ln(1+164)ln10
Now:
ln(1+x) = x-x^2/2+x^3/3-x^4/4+...
So:
ln(1+1/64) = 1/64-1/(2*64^2)+1/(3*64^3)-1/(4*64^4)+...
color(white)(ln(1+1/64)) = 1/64-1/8192+1/786432-1/67108864+...
color(white)(ln(1+1/64)) ~~ 0.01550419
and:
log 1300 = log 10 + 7 log 2 + ln(1+1/64)/ln 10
color(white)(log 1300) ~~ 1 + 7*0.30103 + 0.01550419/2.302585
color(white)(log 1300) ~~ 3.1139434
