What is expand ln(((2pimkT)/(h^2))^{3/2}) ?

The solution to expand ln(((2pimkT)/(h^2))^{3/2}) is 3/2 ln(2pikmT)-3ln(h)

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expand ln(((2pimkT)/(h^2))^{3/2})

expand

ln ((\frac{2 πmk T}{h ^{2}})^{\frac{3}{2}})

\frac{3}{2} ln (2 πkm T) - 3 ln (h)

Solution steps

ln ((\frac{2 πmk T}{h ^{2}})^{\frac{3}{2}})

Apply log rule:

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

ln ((\frac{2 πmk T}{h ^{2}})^{\frac{3}{2}}) = \frac{3}{2} ln (\frac{2 πmk T}{h ^{2}})

= \frac{3}{2} ln (\frac{2 πmk T}{h ^{2}})

Apply log rule:

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

ln (\frac{2 πmk T}{h ^{2}}) = ln (2 πmk T) - ln (h^{2})

= \frac{3}{2} (ln (2 πkm T) - ln (h^{2}))

Apply log rule:

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

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

= \frac{3}{2} (ln (2 πmk T) - 2 ln (h))

= \frac{3}{2} (ln (2 πkm T) - 2 ln (h))

Apply the distributive law:

a (b - c) = ab - a c

a = \frac{3}{2}, b = ln (2 πmk T), c = 2 ln (h)

= \frac{3}{2} ln (2 πmk T) - \frac{3}{2} \cdot 2 ln (h)

= \frac{3}{2} ln (2 πkm T) - 2 \cdot \frac{3}{2} ln (h)

2 \cdot \frac{3}{2} ln (h) = 3 ln (h)

= \frac{3}{2} ln (2 πkm T) - 3 ln (h)

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0.77 + \frac{8.134 \cdot 298}{1 \cdot 96500} \cdot ln (\frac{1}{2})

What is expand ln(((2pimkT)/(h^2))^{3/2}) ?
The solution to expand ln(((2pimkT)/(h^2))^{3/2}) is 3/2 ln(2pikmT)-3ln(h)