Question

If $\log \Big(\frac{x-y}{5}\Big) = \frac{1}{2} \log x+\frac{1}{2} \log y$, show that x² + y² = 27xy.

Answer 1

Answer:

Step-by-step explanation:

Hi,  

We will be using the following properties of    

logarithm:  

Additive Property : logₐx + logₐy = logₐ(xy) ,

Exponent Property : nlogₐx = logₐxⁿ

and log a = log b, then a = b

Given that log ( x - y)/5 = 1/2 *{ log x + log y}

Multiplying by 2 on  both sides , we get  

2log(x - y)/5 = log x + log y  

2log(x - y)/5  = log {(x - y)/5}² [ Using Exponent Property]

log x + log y = log xy [ Using Additive Property]

So, we get  log (x - y)²/25  = log xy

Since logarithms are equal, their numbers should  

be equal  

Hence, (x - y)²/25 = xy

(x - y)² = 25xy

x² + y² - 2xy = 25xy

x² + y² = 27xy

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