Question

How do I find the number whose common logarithm is 2.6025?

Answer 1

Can use `log_10(2) ~= 0.30103` to find approximate value `400` then multiply by an approximation for `10^0.00044 ~= 1.001` to get `400.4`.

If you have a calculator then just `10^(2.6025) ~= 400.4055`

Explanation:

`10^(2.6025) = 10^2*10^0.60206*10^0.00044`

`~=100*10^(2log_10(2))*10^0.00044`

`=100*2^2*10^0.00044`

`=400*10^0.00044`

Now

`1.001^1000 = (1+0.001)^1000`

`=1+((1000),(1))0.001+((1000),(2))0.001^2+...`

`=1+1+0.4995+0.166167+0.04141712475+...`

somewhere between `2.7` and `3`

`log_10(3) ~= 0.4771` (one of those useful numbers to memorise)

`log_10(2.7) = log_10(27/10) = log_10(3^3/10) = 3log_10(3)-1`

`~= 0.4313`

So `0.4313 < 1000 log_10(1.001) < 0.4771`

`0.0004313 < log_10(1.001) < 0.0004771`

So `1.001` is a fairly good approximation for `10^0.00044`