Question

log (x+y/3)=1/2(log x+logy)find the value of x/y+y/x​

Answer 1

Answer:

7

Step-by-step explanation:

The answer is in above image

Answer 2

Question :

if $log( \frac{x + y}{3} ) = \frac{1}{2} ( log(x) + log_{}(y) )$

find the value of

$\frac{ x}{y} + \frac{ y}{x}$

Properties of logarithm :

$1) log(a) + log(b) = log(b)$

$2) log( \frac{a}{b} ) = log(a) - log(b)$

$3) log(a) {}^{n} = n log(a)$

$4) log_{x}(x) = 1$

Solution :

$log( \frac{x + y}{3} ) = \frac{1}{2} ( log(x) + log_{}(y) )$

use property :

log(a) + log (b) = log (ab)

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$log( \frac{x + y}{3} ) = \frac{1}{2} log(xy)$

$log( \frac{x + y}{3} ) - \frac{1}{2} log(xy) = 0$

use property log (a/b) = log a- log b

____________________

$log(x + y) - log(3) - \frac{1}{2} log(xy) = 0$

$2 log(x + y) - log(xy) = 2 log(3)$

use property :

$log(a) {}^{n} = n log(a)$

$log(x + y) {}^{2} - log(xy) = log(3) {}^{2}$

Now use property ;

log (x/y ) = logx -logy

________________________

$log( \frac{(x + y) {}^{2} }{xy} ) = log(9)$

⇒$\frac{(x + y ){}^{2} }{xy} = 9$

$\frac{x {}^{2} + y {}^{2} + 2xy}{xy} = 9$

$\frac{x {}^{2} }{xy} + \frac{y {}^{2} }{xy} + \frac{2xy}{xy} = 9$

$\frac{x}{y} + \frac{y}{x} = 7$

it is the required solution!