What is ln((2^n)/((2+n)^{3/4)}) ?

The solution to ln((2^n)/((2+n)^{3/4)}) is nln(2)-3/4 ln(2+n)

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simplify ln((2^n)/((2+n)^{3/4)})

simplify

ln (\frac{2 ^{n}}{( 2 + n ) ^{\frac{3}{4}}})

n ln (2) - \frac{3}{4} ln (2 + n)

Solution steps

ln (\frac{2 ^{n}}{( 2 + n ) ^{\frac{3}{4}}})

Apply log rule:

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

ln (\frac{2 ^{n}}{( 2 + n ) ^{\frac{3}{4}}}) = ln (2^{n}) - ln ((2 + n)^{\frac{3}{4}})

= ln (2^{n}) - ln ((n + 2)^{\frac{3}{4}})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= n ln (2) - ln ((2 + n)^{\frac{3}{4}})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((2 + n)^{\frac{3}{4}}) = \frac{3}{4} ln (2 + n)

= n ln (2) - \frac{3}{4} ln (2 + n)

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What is ln((2^n)/((2+n)^{3/4)}) ?
The solution to ln((2^n)/((2+n)^{3/4)}) is nln(2)-3/4 ln(2+n)