What is ln(1/x (1/x)^{3/2}) ?

The solution to ln(1/x (1/x)^{3/2}) is -5/2 ln(x)

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simplify ln(1/x (1/x)^{3/2})

simplify

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

- \frac{5}{2} ln (x)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (\frac{1}{x}) = ln (1) - ln (x), ln (\frac{1}{x}) = ln (1) - ln (x)

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

ln (1) = 0

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

= 0 - ln (x) - \frac{3}{2} ln (x)

Add similar elements:

- ln (x) - \frac{3}{2} ln (x) = - \frac{5}{2} ln (x)

= 0 - \frac{5}{2} ln (x)

0 - \frac{5}{2} ln (x) = - \frac{5}{2} ln (x)

= - \frac{5}{2} ln (x)

lo g_{2} (\frac{x}{y ^{3}})

- lo g_{10} (4.5 \cdot 1 0^{- 3})

- lo g_{10} (6.88 \cdot 1 0^{- 4})

lo g_{2} (\frac{a ^{2} - 4}{9})

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