怎樣手算log(3,2)?

05-02

用log(3,2)表示以2為底，3的對數。

以下給出一種手算方法，原理上可以任意精度。但是計算量超級大，有大量形如a^10的乘方計算。供大家參考。

log(3,2)

=1+log(3/2,2)

=1+log(1.5,2)

=1+0.1*log(1.5^10,2)

=1+0.1*log(57.66503906,2)

=1+0.1*(5+log(57.66503906/32,2))

=1+0.1*(5+log(1.802032471,2))

=1+0.5+0.1*log(1.802032471,2)

=1+0.5+0.01*log(1.802032471^10,2)

=1+0.5+0.01*log(361.0988642,2)

=1+0.5+0.01*(8+log(361.0988642/256,2))

=1+0.5+0.01*(8+log(1.410542438,2))

=1+0.5+0.08+0.01*log(1.410542438,2)

=1+0.5+0.08+0.001*log(1.410542438^10,2)

=1+0.5+0.08+0.001*log(31.1789561,2)

=1+0.5+0.08+0.001*(4+log(31.1789561/16,2))

=1+0.5+0.08+0.001*(4+log(1.948684756,2))

=1+0.5+0.08+0.004+0.001*log(1.948684756,2)

=1+0.5+0.08+0.004+0.0001*log(1.948684756^10,2)

=1+0.5+0.08+0.004+0.0001*log(789.6158896,2)

=1+0.5+0.08+0.004+0.0001*(9+log(789.6158896/512,2))

=1+0.5+0.08+0.004+0.0001*(9+log(1.542218534,2))

=1+0.5+0.08+0.004+0.0009+0.0001*log(1.542218534,2)

=1+0.5+0.08+0.004+0.0009+0.00001*log(1.542218534^10,2)

=1+0.5+0.08+0.004+0.0009+0.00001*log(76.1130599,2)

=1+0.5+0.08+0.004+0.0009+0.00001*(6+log(76.1130599/64,2))

=1+0.5+0.08+0.004+0.0009+0.00001*(6+log(1.189266561,2))

=1+0.5+0.08+0.004+0.0009+0.00006+0.00001*log(1.189266561,2)

=1.58496+0.00001*log(1.189266561,2)

≈1.58496

直接用計算器計算的結果是1.584962501，手算的精度是足夠用了。