Compare the following logarithms in base 1/2 without a calculator: log(8) and log(512)
Compare through subtraction : log0.5(8)- log0.5(512) = xUsing logarithm rule in addition/subtraction: loga (b)+loga(c) = loga(b*c) ; loga (b)-loga(c) = loga(b/c) where a,b and c are constants (note both logarithms need to have same base 'a' for this rule to apply)
We get: log0.5(8)- log0.5(512) = log0.5(8/512) = log0.5(1/64)Using the logarithm properties: -loga(1) = 0 for any base 'a' -if base a < 1 the logarithmic function is strictly decreasing if base a > 1 the logarithmic function is strictly increasing
In our case, the base a = 1/2 is inferior to 1, this means the logarithm will be positive for x < 1 and negative for x > 1.We have 1/64 < 1 which implies log0.5(1/64) > 0.Hence :log0.5(8) - log0.5(512) > 0And finally:log0.5(8) > log0.5(512)
