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

1 Answer
Jul 14, 2015

Answer:

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#