Question

Plzz help gahhhhh :\

Answer 1

The product of 'x' and 'y' is
`\boxed 8`.

Explanation:

Given:

`\log_(5\sqrt5)125=x\\\\\log_(2\sqrt2)64=y`

We need to determine the product of 'x' and 'y'.

Using the following logarithmic property:

`\log_ab=(\log b)/(\log a)`

Here,
`a=5\sqrt5\ and\ 2\sqrt2`

`b=125\ and\ 64`

So,
`log_(5\sqrt5)125=(\log 125)/(\log 5√(5))\\\\log_(5\sqrt5)125=(\log 5^3)/(\log 5*5^(1/2)).......[\sqrt5=5^(1/2)]`

`log_(2\sqrt2)64=(\log 64)/(\log 2√(2))\\\\log_(2\sqrt2)64=(\log 2^6)/(\log 2*2^(1/2)).......[\sqrt2=2^(1/2)]`

Now, we use another property of log and exponents.

`\log a^m=m\log a\\a^m* a^n=a^(m+n)`

`log_(5\sqrt5)125=\frac{3\log 5}{\log 5^{1+{1/2}}}=\frac{3\log 5}{\log 5^{(3)/(2)}}=(3\log 5)/((3)/(2)\log 5)=2\\\\\\\\log_(2\sqrt2)64=\frac{6\log 2}{\log 2^{1+{1/2}}}=\frac{6\log 2}{\log 2^{(3)/(2)}}=(6\log 2)/((3)/(2)\log 2)=(12)/(3)=4`

So,
`x=2\ and\ y=4`

The product of 'x' and 'y' =
`2* 4=8`

Therefore, the product of 'x' and 'y' is 8.