Question

Solve the problem.20

Answer 1

Answer:

x = 1/25

y = 1/2

Step-by-step explanation:

Given :

$\sf \dfrac{log \ x}{log \ 5} = \dfrac{log \ {y}^{2} }{log \ 2} = \dfrac{log \ 9}{log \ \frac{1}{3} }$

Using, Base changing formula, log a / log b = log(b) a

$\Rightarrow \sf log_{5}x= log_{2} {y}^{2} = log_{ \frac{1}{3} } {3}^{2}$

Using a^m = 1 / a^( - m )

$\Rightarrow \sf log_{5}x= log_{2} {y}^{2} = log_{ \frac{1}{3} } \dfrac{1}{3^{ - 2} }$

$\Rightarrow \sf log_{5}x= log_{2} {y}^{2} = log_{ \frac{1}{3} } \bigg(\dfrac{1}{3} \bigg)^{ - 2}$

Using Power rule log a^m = m.log a

$\Rightarrow \sf log_{5}x= log_{2} {y}^{2} = - 2 log_{ \frac{1}{3} } \dfrac{1}{3}$

$\Rightarrow \sf log_{5}x= log_{2} {y}^{2} = - 2$

$\Rightarrow \sf log_{5}x= - 2 \qquad log_{2} {y}^{2} = - 2$

Writing it in exponential form

$\Rightarrow \sf 5^{ - 2} = x \qquad {2}^{ - 2} = {y}^{2}$

$\Rightarrow \sf \dfrac{1}{ {5}^{2} } = x \qquad {2}^{ - 1} = y$

$\Rightarrow \sf x = \dfrac{1}{25} \qquad y = \dfrac{1}{2}$

Therefore the value of x is 1/25 and value of y is 1/2.