What is 4ln(2)-4ln(2x) ?

The solution to 4ln(2)-4ln(2x) is -ln(x^4)

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simplify 4ln(2)-4ln(2x)

simplify

4 ln (2) - 4 ln (2 x)

- ln (x^{4})

Solution steps

4 ln (2) - 4 ln (2 x)

Apply log rule:

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

4 ln (2) = ln (2^{4})

= ln (2^{4}) - 4 ln (2 x)

Apply log rule:

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

4 ln (2 x) = ln ((2 x)^{4})

= ln (2^{4}) - ln ((2 x)^{4})

Apply log rule:

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

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

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

Simplify

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

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

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (x^{4})

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