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konrad509

[tex]xy=\log_{5\sqrt5}{125}\cdot\log_{2\sqrt2}64\\\\xy=\dfrac{\log_5125}{\log_5 5\sqrt5}\cdot\dfrac{\log_264}{\log_22\sqrt2}\\\\xy=\dfrac{3}{\log_55^{\tfrac{3}{2}}}\cdot\dfrac{6}{\log_22^{\tfrac{3}{2}}}\\\\xy=\dfrac{3}{\dfrac{3}{2}}\cdot\dfrac{6}{\dfrac{3}{2}}\\\\xy=3\cdot\dfrac{2}{3}\cdot6\cdot\dfrac{3}{2}\\\\xy=18[/tex]

Answer:

The product of x and y is 8.

Step-by-step explanation:

It is given that

[tex]x=\log_{5\sqrt{5}}\left(125\right)[/tex]

[tex]y=\log_{2\sqrt{2}}\left(64\right)[/tex]

We need to find the product of x and y.

[tex]x\cdot y=\log_{5\sqrt{5}}\left(125\right)\cdot\log_{2\sqrt{2}}\left(64\right)[/tex]

It can be written as

[tex]xy=\log_{5\sqrt{5}}\left(5\sqrt{5}\right)^2\cdot\log_{2\sqrt{2}}\left(2\sqrt{2}\right)^4[/tex]

Using the properties of logarithm, we get

[tex]xy=2\cdot 4[/tex]                    [tex][\because log_aa^x=x][/tex]

[tex]xy=8[/tex]

Therefore the product of x and y is 8.

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