2023/017) Simplify the following log expression

$log(10+3\sqrt{10}) + log(10+\sqrt{90+\sqrt{90}})+ log(10- \sqrt{90+\sqrt{90}})$

Solution:

Add the 2nd and 3rd expression and knowing $log\, a + log\, b = log\, ab$

  $log(10+\sqrt{90+\sqrt{90}})+ log(10- \sqrt{90+\sqrt{90}})$

  $= log((10+\sqrt{90+\sqrt{90}})(10-\sqrt{90+\sqrt{90}}))$

$= log(10^2 - (\sqrt{90 +\sqrt{90}})^2$ using $(a+b)(a-b) = a^ 2- b^2$

$=log( 100 - (90 + \sqrt{90}))$

$= log(100 - 90 - \sqrt{90})$

$= log (10 - \sqrt{3^2 * 10})$ factoring 90 to product of a square and no square

$= log( 10 - 3 \sqrt{10})$

So given expression = $log(10+3\sqrt{10})  +  log( 10 - 3 \sqrt{10})$

$= log ( (10+3\sqrt{10})(10- 3\sqrt{10}))$ knowing $log\, a + log\, b = log\, ab$

$=log (10^2 - (3\sqrt{10})$

$= log (100 - 9 * 10)$

$=log\, 10 = 1$