What is 4ln(y/x)+4ln(x) ?

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

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simplify 4ln(y/x)+4ln(x)

simplify

4 ln (\frac{y}{x}) + 4 ln (x)

ln (y^{4})

Solution steps

4 ln (\frac{y}{x}) + 4 ln (x)

Apply log rule:

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

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

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

Apply log rule:

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

4 ln (x) = ln (x^{4})

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

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

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

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

Simplify

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

= ln (y^{4})

ln (y = x \cdot e^{x^{2}} \cdot (x^{2} + 10)^{10})

ln (ln (5) - 7 ln (10))

ln (\frac{2000}{1999}) 111

- 0.402 - \frac{0.05916}{2} lo g_{10} (\frac{0.5}{1})

4.184 \cdot ln (\frac{( 273.15 + 60 )}{( 273.15 + 18 )})