Question Number 101498 by bemath last updated on 03/Jul/20

If 4^x  = 5^y  = 20^z  .  what is z in term of x and y.  ★♠

Commented bybobhans last updated on 03/Jul/20

x ln(4) = y ln(5) = z (ln(4)+ln(5))  ⇒(1) ln(4) = ((y ln(5))/x)  ⇒(2) z = ((y ln(5))/(ln(4)+ln(5))) = ((y ln(5))/(((y ln(5))/x) + ln(5)))       z = ((yx)/(y+x)) = ((xy)/(x+y)) ♥◊

Commented byDwaipayan Shikari last updated on 03/Jul/20

4^x =5^y =20^z =k  (k≠0)  4=k^(1/x)      5=k^(1/y)      20=k^(1/z)   k^(1/x) k^(1/y) =k^(1/z)   (1/z)=(1/x)+(1/y)    ⇒z=((1/x)+(1/y))^(−1) =((xy)/(x+y))  ■ ★ L

Commented bybramlex last updated on 03/Jul/20

thank you both

Answered by RKT last updated on 03/Jul/20

4^x =20^z   ⇒4^x =4^z ×5^z   ⇒4^(x−z) =5^z   ⇒4^((x−z)/z) =5  ⇒4^((y/z)(x−z)) =5^y   ⇒4^((y/z)(x−z)) =4^x ∵4^x =5^y   ⇒(y/z)(x−z)=x  ⇒xy−yz=zx  ⇒xy=zx+yz  ⇒xy=z(x+y)  ⇒((xy)/(x+y))=z