What is 2ln(x)-2ln(y/x) ?

The solution to 2ln(x)-2ln(y/x) is ln((x^4)/(y^2))

Popular >

simplify 2ln(x)-2ln(y/x)

simplify

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

ln (\frac{x ^{4}}{y ^{2}})

Solution steps

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

Apply log rule:

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

2 ln (x) = ln (x^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

\frac{x ^{2}}{( \frac{y}{x} ) ^{2}} : \frac{x ^{4}}{y ^{2}}

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

\frac{1}{2} (ln (61) - ln (6))

ln (y - 4) + ln (y + 4)

- lo g_{10} (8.5 \cdot 1 0^{- 3})

4 ln (- 2) + ln (- 3)

lo g_{5} (\frac{1}{40})