What is 23/2 (2*ln((a^3)/(sqrt(b)))+22*ln(1/b)) ?

The solution to 23/2 (2*ln((a^3)/(sqrt(b)))+22*ln(1/b)) is 23/2 ln((a^6)/(b^{23)})

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simplify 23/2 (2ln((a^3)/(sqrt(b)))+22ln(1/b))

simplify

\frac{23}{2} (2 \cdot ln (\frac{a ^{3}}{b}) + 22 \cdot ln (\frac{1}{b}))

\frac{23}{2} ln (\frac{a ^{6}}{b ^{23}})

Solution steps

\frac{23}{2} (2 ln (\frac{a ^{3}}{b}) + 22 ln (\frac{1}{b}))

Apply log rule:

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

2 ln (\frac{a ^{3}}{b}) = ln ((\frac{a ^{3}}{b})^{2})

= \frac{23}{2} (22 ln (\frac{1}{b}) + ln ((\frac{a ^{3}}{b})^{2}))

Apply log rule:

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

22 ln (\frac{1}{b}) = ln ((\frac{1}{b})^{22})

= \frac{23}{2} (ln ((\frac{1}{b})^{22}) + ln ((\frac{a ^{3}}{b})^{2}))

Apply log rule:

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

ln ((\frac{a ^{3}}{b})^{2}) + ln ((\frac{1}{b})^{22}) = ln ((\frac{1}{b})^{22} (\frac{a ^{3}}{b})^{2})

= \frac{23}{2} ln ((\frac{1}{b})^{22} (\frac{a ^{3}}{b})^{2})

Simplify

(\frac{a ^{3}}{b})^{2} (\frac{1}{b})^{22} : \frac{a ^{6}}{b ^{23}}

= \frac{23}{2} ln (\frac{a ^{6}}{b ^{23}})

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What is 23/2 (2*ln((a^3)/(sqrt(b)))+22*ln(1/b)) ?
The solution to 23/2 (2*ln((a^3)/(sqrt(b)))+22*ln(1/b)) is 23/2 ln((a^6)/(b^{23)})