What is 1/2 ln(3c)+1/2 ln(4)d-2ln(5e) ?

The solution to 1/2 ln(3c)+1/2 ln(4)d-2ln(5e) is ln((sqrt(3)sqrt(c))/(25e^2))+ln(2)d

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simplify 1/2 ln(3c)+1/2 ln(4)d-2ln(5e)

simplify

\frac{1}{2} ln (3 c) + \frac{1}{2} ln (4) d - 2 ln (5 e)

ln (\frac{3 c}{25 e ^{2}}) + ln (2) d

Solution steps

\frac{1}{2} ln (3 c) + \frac{1}{2} ln (4) d - 2 ln (5 e)

Apply log rule:

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

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

= ln ((3 c)^{\frac{1}{2}}) + \frac{1}{2} ln (4) d - 2 ln (5 e)

Apply log rule:

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

2 ln (5 e) = ln ((5 e)^{2})

= ln ((3 c)^{\frac{1}{2}}) + \frac{1}{2} ln (4) d - ln ((5 e)^{2})

Apply log rule:

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

ln ((3 c)^{\frac{1}{2}}) - ln ((5 e)^{2}) = ln (\frac{( 3 c ) ^{\frac{1}{2}}}{( 5 e ) ^{2}})

= ln (\frac{( 3 c ) ^{\frac{1}{2}}}{( 5 e ) ^{2}}) + \frac{1}{2} ln (4) d

Simplify

\frac{( 3 c ) ^{\frac{1}{2}}}{( 5 e ) ^{2}} : \frac{3 c}{25 e ^{2}}

= ln (\frac{3 c}{25 e ^{2}}) + \frac{1}{2} ln (4) d

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

= ln (\frac{3 c}{25 e ^{2}}) + ln (4) \frac{d}{2}

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

= ln (\frac{3 c}{25 e ^{2}}) + ln (2) d

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What is 1/2 ln(3c)+1/2 ln(4)d-2ln(5e) ?
The solution to 1/2 ln(3c)+1/2 ln(4)d-2ln(5e) is ln((sqrt(3)sqrt(c))/(25e^2))+ln(2)d