Question

Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator.
log4(2x2 - 20x + 12y)

Answer 1

Answer:

$f(x,y) = \log_{4} (x-5-\sqrt{25-6\cdot y})+\log_{4} (x-5+\sqrt{25-6\cdot y})$

Step-by-step explanation:

Let be $f(x,y) = \log_{4}(2\cdot x^{2}-20\cdot x +12\cdot y)$, this expression is simplified by algebraic and trascendental means. As first step, the second order polynomial is simplified. Its roots are determined by the Quadratic Formula, that is to say:

$r_{1,2} = \frac{20\pm \sqrt{(-20)^{2}-4\cdot (2)\cdot (12\cdot y)}}{2\cdot (2)}$

$r_{1,2} = 5\pm \sqrt{25-6\cdot y}$

The polynomial in factorized form is:

$(x-5-\sqrt{25-6\cdot y})\cdot (x-5+\sqrt{25-6\cdot y})$

The function can be rewritten and simplified as follows:

$f(x,y) = \log_{4} [(x-5-\sqrt{25-6\cdot y})\cdot (x-5+\sqrt{25-6\cdot y})]$

$f(x,y) = \log_{4} (x-5-\sqrt{25-6\cdot y})+\log_{4} (x-5+\sqrt{25-6\cdot y})$