Question

5^x=(0.5)^y=1000 then (y-x)(x+y)

Answer 1

I think your question is --> 5^x = (0.5)^y = 1000 find (y - x)/(xy)= ?

it is given that , 5^x = (0.5)^y = 1000

now 5^x = 1000 = (10)³

taking log base 10 both sides,

⇒ log10 (5^x) = log10 (10³)

⇒$xlog_{10}5=3log_{10}10$

⇒$xlog_{10}5=3$ [as we know, $log_{10}10=1$ ]

⇒$x=\frac{3}{log_{10}5}$.....(1)

similarly, (0.5)^y = 1000

⇒ (1/2)^y = 10³

⇒2^-y = 10³

taking log base 10 both sides,

⇒$-ylog_{10}2=3$

⇒y=$\frac{-3}{log_{10}2}$

now, (y-x) = $\frac{-3}{log_{10}2}-\frac{3}{log_{10}5}$

= $-3\frac{log_{10}2+log_{10}5}{log_{10}5.log_{10}2}=-3\frac{log_{10}2\times5}{log_{10}5.log_{10}2}$

= $\frac{-3}{log_{10}5.log_{10}2}$....(1)

and xy = $\frac{3\times-3}{log_{10}5.log_{10}2}$

=$\frac{-9}{log_{10}5.log_{10}2}$.....(2)

now, from equations (1) and (2),

(y - x)/xy = $\frac{-3}{-9}$

= 1/3

hence, (y - x)/xy = 1/3

Answer 2

it is given that , 5^x = (0.5)^y = 1000

now 5^x = 1000 = (10)³

taking log base 10 both sides,

⇒ log10 (5^x) = log10 (10³)

⇒xlog_{10}5=3log_{10}10xlog

10

5=3log

10

10

⇒xlog_{10}5=3xlog

10

5=3 [as we know, log_{10}10=1log

10

10=1 ]

⇒x=\frac{3}{log_{10}5}x=

log

10

5

3

.....(1)

similarly, (0.5)^y = 1000

⇒ (1/2)^y = 10³

⇒2^-y = 10³

taking log base 10 both sides,

⇒-ylog_{10}2=3−ylog

10

2=3

⇒y=\frac{-3}{log_{10}2}

log

10

2

−3

now, (y-x) = \frac{-3}{log_{10}2}-\frac{3}{log_{10}5}

log

10

2

−3

−

log

10

5

3

= -3\frac{log_{10}2+log_{10}5}{log_{10}5.log_{10}2}=-3\frac{log_{10}2\times5}{log_{10}5.log_{10}2}−3

log

10

5.log

10

2

log

10

2+log

10

5

=−3

log

10

5.log

10

2

log

10

2×5

= \frac{-3}{log_{10}5.log_{10}2}

log

10

5.log

10

2

−3

....(1)

and xy = \frac{3\times-3}{log_{10}5.log_{10}2}

log

10

5.log

10

2

3×−3

=\frac{-9}{log_{10}5.log_{10}2}

log

10

5.log

10

2

−9

.....(2)

now, from equations (1) and (2),

(y - x)/xy = \frac{-3}{-9}

−9

−3

= 1/3

hence, (y - x)/xy = 1/3