What is ln((x^{1/2})/(x^2sqrt(1+x^2))) ?

The solution to ln((x^{1/2})/(x^2sqrt(1+x^2))) is -3/2 ln(x)-ln(sqrt(1+x^2))

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simplify ln((x^{1/2})/(x^2sqrt(1+x^2)))

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

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

Apply log rule:

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

ln (x^{2} 1 + x^{2}) = ln (x^{2}) + ln (1 + x^{2})

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

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

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

- (2 ln (x) + ln (1 + x^{2})) : - 2 ln (x) - ln (1 + x^{2})

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

Add similar elements:

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

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

\frac{1}{2} ln (3 x - 5 y) - ln (4 x + y)

lo g_{b} (7 x \cdot 3 y)

lo g_{2} (5) + \frac{1}{2} lo g_{2} (36) - lo g_{2} (10)

lo g_{4} (16 y z^{3})

lo g_{10} (5 x + 8) - lo g_{10} (x)

What is ln((x^{1/2})/(x^2sqrt(1+x^2))) ?
The solution to ln((x^{1/2})/(x^2sqrt(1+x^2))) is -3/2 ln(x)-ln(sqrt(1+x^2))