if a^x=(a/k)^y=k^m and a is not equal to 1 then show that 1/x-1/y=1/m

Question

show me work plz

Answer 1

(a/k)^y=k^m
take log of both sides:
y(log (a/k) = mlogk
y(log a - logk = mlogk
yloga = mlogk + ylogk
yloga = logk (m+y)
logk = yloga/(m+y)

also:
a^x = k^m
xloga = mlogk
logk = xloga/m

xloga /m = yloga/(m+y)
divide by loga
x/m = y/(m+y)
ym = x(m+y)
ym = xm + xy
ym - xm = xy
m(y-x) = xy
m = xy/(y-x)
m = x(x+y)/y

we are supposed to show that 1/x - 1/y = 1/m
1/m = (y-x)/(xy)
m = xy/(y-x), which is what I have done above.

checking:
let y = 6, x = 4, then m = 12

and if a^x = k^m
a^4 = k^12
4loga = 12logk
log a = 3logk
log a = log k^3
a = k^3, let k = 2, then a = 8

checking the question:
a^x=(a/k)^y=k^m
8^4 = 4096
(a/k)^y =(8/2)^6 = 4^6 = 4096
k^m = 2^12 = 4096

but
1/4 - 1/6 = 1/12, illustrating the result

Answer 2

Thankyou somuch

Answer 3

Thank you so much