Question

Plzz help gahhhhh :\

Answer 1

Answer:

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

Step-by-step 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:

$\log_ab=\frac{\log b}{\log a}$

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

$b=125\ and\ 64$

So, $log_{5\sqrt5}125=\frac{\log 125}{\log 5\sqrt{5}}\\\\log_{5\sqrt5}125=\frac{\log 5^3}{\log 5\times5^{1/2}}.......[\sqrt5=5^{1/2}]$

$log_{2\sqrt2}64=\frac{\log 64}{\log 2\sqrt{2}}\\\\log_{2\sqrt2}64=\frac{\log 2^6}{\log 2\times2^{1/2}}.......[\sqrt2=2^{1/2}]$

Now, we use another property of log and exponents.

$\log a^m=m\log a\\a^m\times a^n=a^{m+n}$

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

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

The product of 'x' and 'y' = $2\times 4=8$

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