Express (In 35+ln (1/7)) / In 25 in terms of In 5 and In 7

Properties of the logarithm: for any base of logarithm,

log (a*b) = log (a) + log (b)

If we replace b with 1/b, or b^-1, we have

log (a/b) = log (a) + log (1/b) = log (a) - log (b)

since

log (1/b) = log (b^-1) = - log (b)

using the power property of logarithms,

log (b^n) = n log (b)

Now,

ln35 = ln (5*7) = ln5 + ln7

ln (1/7) = - ln7

ln25 = ln (5^2) = 2 ln5

Putting everything together, we have

(ln35 + ln (1/7)) / ln25 = (ln5 + ln7 - ln7) / (2 ln5) = ln5 / (2 ln5) = 1/2