What is ln((sqrt(2))/((3/2)^{1/4)}) ?

The solution to ln((sqrt(2))/((3/2)^{1/4)}) is -1/4 ln(3)+3/4 ln(2)

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simplify ln((sqrt(2))/((3/2)^{1/4)})

simplify

ln \frac{2}{( \frac{3}{2} ) ^{\frac{1}{4}}}

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

Decimal Notation

0.24520 \dots

Solution steps

ln \frac{2}{( \frac{3}{2} ) ^{\frac{1}{4}}}

Apply log rule:

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

ln \frac{2}{( \frac{3}{2} ) ^{\frac{1}{4}}} = ln (2) - ln ((\frac{3}{2})^{\frac{1}{4}})

= ln (2) - ln ((\frac{3}{2})^{\frac{1}{4}})

Apply log rule:

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

ln ((\frac{3}{2})^{\frac{1}{4}}) = \frac{1}{4} ln (\frac{3}{2})

= ln (2) - \frac{1}{4} ln (\frac{3}{2})

Apply log rule:

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

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

= ln (2) - \frac{1}{4} (ln (3) - ln (2))

Apply log rule

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

assuming

x \geq 0

= \frac{1}{2} ln (2) - \frac{1}{4} (ln (3) - ln (2))

Expand

- \frac{1}{4} (ln (3) - ln (2)) : - \frac{1}{4} ln (3) + \frac{1}{4} ln (2)

= \frac{1}{2} ln (2) - \frac{1}{4} ln (3) + \frac{1}{4} ln (2)

Simplify

\frac{1}{2} ln (2) - \frac{1}{4} ln (3) + \frac{1}{4} ln (2) : - \frac{1}{4} ln (3) + \frac{3}{4} ln (2)

= - \frac{1}{4} ln (3) + \frac{3}{4} ln (2)

3 lo g_{3} (x) + 9 lo g_{3} (z)

lo g_{\frac{1}{2}} (\frac{8}{15})

5 ln (x) + 4 ln (y) - 3 ln (z)

lo g_{6} ((\frac{x}{216})^{4})

What is ln((sqrt(2))/((3/2)^{1/4)}) ?
The solution to ln((sqrt(2))/((3/2)^{1/4)}) is -1/4 ln(3)+3/4 ln(2)