Question

Logb(base a)=3/2 log d(base c)=5/7 and a-c=9 then b-d=?

Answer 1

log_a b = 3/2 ----> a^(3/2) = b or a^3 = b^2

log_c d = 5/7 ----> c^(5/7) = d or c^5 = d^7
we also know that a-c = 9

Intuitively , if a^3 = b^2, then b>a, One value that would work is a=4, b=8
then in a-c = 9, c = a-9 , a>9 or else the logs are not defined


ok, let's assign some values:
let a = 12, then 12^3 = b^2, b = √1728 = appr 41.569
c = 12 - 9 = 3
d^7 = c^5 , on my calculator I get d = appr 2.1918

check: log₁₂ 41.569 = 1.5
log₃ 2.1918 = .71428... and 5/7 = .71428..
and a-c = 12-3 = 9

so one solution is:
a = 12
b = appr 41.569
c = 3
d = appr 2.1918 and in this case
b-d = 39.377

Using the above method you could find an infinite number of solutions and show
that b-d is not unique