How do you calculate #log 10.000001# ?

1 Answer
George C.
Sep 16, 2017

#log 10.000001 ~~ 1.00000004342945#

Explanation:

Use:

#ln 10 ~~ 2.302585093#

#ln(1+t) = t-t^2/2+t^3/3-t^4/4+...#

#log_b a = log_c a / log_c b" "# (Change of base formula)

Then:

#log 10.000001 = log(10*1.0000001)#

#color(white)(log 10.000001) = log(10) + log(1+0.0000001)#

#color(white)(log 10.000001) = 1 + ln(1+0.0000001)/ln(10)#

#color(white)(log 10.000001) = 1 + 1/ln(10)(0.0000001-0.0000001^2/2+0.0000001^3/3-...)#

#color(white)(log 10.000001) ~~ 1 + 0.0000001/ln(10)#

#color(white)(log 10.000001) ~~ 1 + 0.0000001/2.302585093#

#color(white)(log 10.000001) ~~ 1.00000004342945#

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