Answer:
[tex]x\cdot y=8[/tex]
Step-by-step explanation:
Exponential Equations
They can be solved by normal algebraic techniques, provided both sides of the equations are powers of the same base.
[tex]log_{5\sqrt{5}}125=x[/tex]
The log properties state that:
[tex]log_xy=z=>x^z=y[/tex]
Applying to the given equation
[tex](5\sqrt{5})^x=125[/tex]
Transforming both sides as a power of 5:
[tex]\displaystyle 5^{\frac{3}{2}x}=5^3[/tex]
Simplifying
[tex]\displaystyle \frac{3}{2}x=3[/tex]
Solving for x
[tex]x=2[/tex]
Now for the second equation
[tex]log_{2\sqrt{2}}64=y[/tex]
Applying the log property
[tex](2\sqrt{2})^y=64[/tex]
[tex]\displaystyle 2^{\frac{3}{2}y}=2^6[/tex]
Simplifying
[tex]\displaystyle \frac{3}{2}y=6[/tex]
Solving for y:
[tex]y=4[/tex]
The product of x and y is
[tex]xy=(2)(4)=8[/tex]
