Relation between $x,y,z$....Exponent problem...

Question

The given equation is-

$\sqrt[x]{75} = \sqrt[y]{45} =\sqrt[z]{15}$

Now,it is required to prove $x+y=3z$.

I want the simplest possible solution.Thanks in advance.

Answer 1

From the statement of the problem, we know that $$15^{y/z}=45$$ and $$15^{x/z}=75$$

Consider $$15^3=(45)(75)=(15^{y/z})(15^{x/z})=15^{\frac{x+y}{z}}$$

From properties of logarithms and exponents, we now have $$\frac{x+y}{z}=3$$ and the result is immediate from here.

Answer 2

Since $$75^{\frac 1x}=15^{\frac 1z}\Rightarrow 75=15^{\frac xz}$$and$$45^{\frac 1y}=15^{\frac 1z}\Rightarrow 45=15^{\frac yz}$$ we have $$15^3=75\times 45=15^{\frac xz}\times 15^{\frac yz}=15^{\frac xz+\frac yz}\Rightarrow 3=\frac xz+\frac yz.$$

Answer 3

seperating them by pairs we get: $$ \frac{1}{x}ln(75)=\frac{1}{z}ln(15) $$ and $$ \frac{1}{y}ln(45)=\frac{1}{z}ln(15) $$ We can then use the fact that $ln(a*b)=ln(a)*ln(b)$ to show: $$ \frac{ln(3)+2ln(5)}{x}=\frac{ln(5)+ln(3)}{z} $$ and $$ \frac{2ln(3)+ln(5)}{y}=\frac{ln(5)+ln(3)}{z} $$ Rearranging gives: $$ ({ln(3)+2ln(5)}){z}=({ln(5)+ln(3)}){x} $$ and $$ ({2ln(3)+ln(5)}){z}=({ln(5)+ln(3)}){y} $$ We can then observe that by summating these simultaneously we get $$ ({3ln(3)+3ln(5)}){z}=({ln(5)+ln(3)})({y+x}) $$ and thus dividing through by $(ln(5)+ln(3))$ we find that $$ x+y=3z $$