Compare the following logarithms in base 1/2 without a calculator: log(8) and log(512)

Compare through subtraction : log_0.5(8)- log_0.5(512) = xUsing logarithm rule in addition/subtraction: log_a (b)+log_a(c) = log_a(b*c) ; log_a (b)-log_a(c) = log_a(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: log_0.5(8)- log_0.5(512) = log_0.5(8/512) = log_0.5(1/64)Using the logarithm properties: -log_a(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 log_0.5(1/64) > 0.Hence :log_0.5(8) - log_0.5(512) > 0And finally:log_0.5(8) > log_0.5(512)