What numbers would the base-10 logarithm of 270 fall between?

1 Answer
Parth K.
Oct 7, 2014

Between 2 and 3.

#\log_10(270) #
#= \log_10(2.7 \times 10^2)#
#= \log_10 (2.7) + \log_10(10^2)#
#=\log_10(2.7) + 2#

Now, #\log(2.7)# lies somewhere between 0 and 1 because 2.7 lies somewhere between #10^0# and #10^1#. In mathematical terms, #10^0 < 2.7 < 10^1 \iff \log(10^0) < \log(2.7) < \log(10^1)#. Therefore #2 + \log(2.7)# must lie between #2 + 0# and #2 + 1# which is #2# and #3#.

Similarly, you can directly state it this way:

#10^2 < 270 < 10^3 #
#\iff \log(10^2) < \log(270) < \log(10^3)#
#\iff 2 < \log(270) < 3#

The latter seems to work better. I just introduced the former because it forces a student to actually think of it in terms of the orders of magnitude.

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