What is 1/4 ln(x^{12})-3ln(2) ?

The solution to 1/4 ln(x^{12})-3ln(2) is ln((x^3)/8)

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simplify 1/4 ln(x^{12})-3ln(2)

simplify

\frac{1}{4} ln (x^{12}) - 3 ln (2)

ln (\frac{x ^{3}}{8})

Solution steps

\frac{1}{4} ln (x^{12}) - 3 ln (2)

Apply log rule:

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

\frac{1}{4} ln (x^{12}) = ln ((x^{12})^{\frac{1}{4}})

= ln ((x^{12})^{\frac{1}{4}}) - 3 ln (2)

Apply log rule:

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

3 ln (2) = ln (2^{3})

= ln ((x^{12})^{\frac{1}{4}}) - ln (2^{3})

Apply log rule:

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

ln ((x^{12})^{\frac{1}{4}}) - ln (2^{3}) = ln \frac{( x ^{12} ) ^{\frac{1}{4}}}{2 ^{3}}

= ln \frac{( x ^{12} ) ^{\frac{1}{4}}}{2 ^{3}}

Simplify

\frac{( x ^{12} ) ^{\frac{1}{4}}}{2 ^{3}} : \frac{x ^{3}}{8}

= ln (\frac{x ^{3}}{8})

lo g_{10} (\frac{a ^{2}}{b ^{2} c})

lo g_{2} (5 \cdot 1 0^{9} \cdot 365 \cdot 24 \cdot 60 \cdot 60 \cdot 1 0^{6})

10 lo g_{10} (\frac{4 π \cdot 0.4}{( 0.075 ) ^{2}})

(5 (ln (5) - ln (2)))^{2} + (ln (5) - ln (2))^{2}

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

What is 1/4 ln(x^{12})-3ln(2) ?
The solution to 1/4 ln(x^{12})-3ln(2) is ln((x^3)/8)