Question

Show that if
log_b (a) = c, and log_y (b) = c, then log_a (y)=c^-2

Answer 1

recall
log_b a = log a/log b (base 10, or any other base for that matter)

so log_b(a) = loga/logb = c
and log_y(b) = logb/logy = c

then [loga/logb][logb/logy] = (c)(c) = c^2
so loga/logy = c^2 or
logy/loga = 1/c^2 = c^-2

but logy/loga = log_a(y)

so log_a(y) = c^-2 as required.