What is 1/2 ln(4)-4ln(5d) ?

The solution to 1/2 ln(4)-4ln(5d) is ln(2/(625d^4))

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simplify 1/2 ln(4)-4ln(5d)

simplify

\frac{1}{2} ln (4) - 4 ln (5 d)

ln (\frac{2}{625 d ^{4}})

Solution steps

\frac{1}{2} ln (4) - 4 ln (5 d)

Apply log rule:

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

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

= ln (4^{\frac{1}{2}}) - 4 ln (5 d)

Apply log rule:

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

4 ln (5 d) = ln ((5 d)^{4})

= ln (4^{\frac{1}{2}}) - ln ((5 d)^{4})

Apply log rule:

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

ln (4^{\frac{1}{2}}) - ln ((5 d)^{4}) = ln (\frac{4 ^{\frac{1}{2}}}{( 5 d ) ^{4}})

= ln (\frac{4 ^{\frac{1}{2}}}{( 5 d ) ^{4}})

Simplify

\frac{4 ^{\frac{1}{2}}}{( 5 d ) ^{4}} : \frac{2}{625 d ^{4}}

= ln (\frac{2}{625 d ^{4}})

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