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

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Answer 1 · 2015-07-14T01:49:31

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 $10^(2.6025) ~= 400.4055$

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

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

$=100*2^2*10^0.00044$

$=400*10^0.00044$

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 $10^0.00044$