Question

Find the value of y

Answer 1

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Answer 2

Concept:

We need know few logarithmic formulas:

$log_{x}x= 1\\logx^{m} =m\ logx\\log_xy.log_ax=log_ay\\log_xy=\frac{log_ay}{log_ax}$

Given:

The following equation is given,

$Let\ , \ L={{5^{log_{5}x} }^{log_{x}y}} ^{log_{z}z^{2} }$               --------(1)

To find:

We need to find the value of y.

Solution:

Now, we see that

$log_{z}z^{2}=2log_{z}z=2$

Substituiting in (1)

${{5^{log_{5}x} }^{log_{x}y}} ^{2 }$                    ---------(2)

Now,

$log_{x}y^{2} =2log_xy$

Now, substituting this value in (2)

${{5^{2log_{x}y log_{5}x}$                   --------(3)

Next,

${2log_{x}y}}\ {log_{5}x} }=2log_5y$

Substituting this in (3)

${{5^{2log_5y}$                            --------------(4)

Therefore, let the equation be L

$L={{5^{2log_5y}$

taking log base 5 on both side,

$log_5L=2log_5y$

⇒$\frac{log_5L}{log_5y} =2$

⇒$log_yL=2$

⇒$y^{2}=L$

⇒$y=\sqrt{L}$

Therefore value of y is root times the given equation.