What is 1/2 (ln(8x^2)-ln(4)-3ln(x)-ln(y)) ?

The solution to 1/2 (ln(8x^2)-ln(4)-3ln(x)-ln(y)) is 1/2 ln(2/(xy))

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simplify 1/2 (ln(8x^2)-ln(4)-3ln(x)-ln(y))

simplify

\frac{1}{2} (ln (8 x^{2}) - ln (4) - 3 ln (x) - ln (y))

\frac{1}{2} ln (\frac{2}{x y})

Solution steps

\frac{1}{2} (ln (8 x^{2}) - ln (4) - 3 ln (x) - ln (y))

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (8 x^{2}) - ln (4) = ln (\frac{8 x ^{2}}{4})

= \frac{1}{2} (ln (\frac{8 x ^{2}}{4}) - 3 ln (x) - ln (y))

Divide the numbers:

\frac{8}{4} = 2

= \frac{1}{2} (ln (2 x^{2}) - 3 ln (x) - ln (y))

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

3 ln (x) = ln (x^{3})

= \frac{1}{2} (ln (2 x^{2}) - ln (x^{3}) - ln (y))

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (2 x^{2}) - ln (x^{3}) = ln (\frac{2 x ^{2}}{x ^{3}})

= \frac{1}{2} (ln (\frac{2 x ^{2}}{x ^{3}}) - ln (y))

Simplify

\frac{2 x ^{2}}{x ^{3}} : \frac{2}{x}

= \frac{1}{2} (ln (\frac{2}{x}) - ln (y))

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (\frac{2}{x}) - ln (y) = ln (\frac{\frac{2}{x}}{y})

= \frac{1}{2} ln (\frac{\frac{2}{x}}{y})

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{1}{2} ln (\frac{2}{x y})

0.53 - (\frac{0.0592}{2}) \cdot lo g_{10} (\frac{0.1}{3})

ln (\frac{24 2}{2})

- lo g_{10} (5.45 \cdot 1 0^{- 5})

lo g_{5} (\frac{2 x ^{3}}{4 y})

\frac{3}{2} + ln (2) - ln (3) + 2 ln (3) - 3 ln (2)

What is 1/2 (ln(8x^2)-ln(4)-3ln(x)-ln(y)) ?
The solution to 1/2 (ln(8x^2)-ln(4)-3ln(x)-ln(y)) is 1/2 ln(2/(xy))