化简 ln(x)+1/2 ln((x^2+y^2)/(x^2))

解答

化简

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

ln (x^{2} + y^{2})

求解步骤

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

使用对数计算法则:

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

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

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

使用对数计算法则:

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

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

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

化简

x (\frac{x ^{2} + y ^{2}}{x ^{2}})^{\frac{1}{2}} : x^{2} + y^{2}

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

s im pl i f y ln (\frac{2}{x ^{2} + 6})

s im pl i f y ln (\frac{1}{2} x^{0.5} \cdot y^{0.5})

e x p an d lo g_{10} (\frac{x ^{5}}{( x + 6 ) ^{3}})

s im pl i f y 3 lo g_{b} (x) - 4 lo g_{b} (y) + \frac{1}{2} lo g_{b} (x + y)

s im pl i f y ln (\frac{10}{8})