What is 4ln(1/(2^k))-4k+4ln(2)k^2-8k^2 ?

The solution to 4ln(1/(2^k))-4k+4ln(2)k^2-8k^2 is -4ln(2)k-4k+4ln(2)k^2-8k^2

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simplify 4ln(1/(2^k))-4k+4ln(2)k^2-8k^2

simplify

4 ln (\frac{1}{2 ^{k}}) - 4 k + 4 ln (2) k^{2} - 8 k^{2}

- 4 ln (2) k - 4 k + 4 ln (2) k^{2} - 8 k^{2}

Solution steps

4 ln (\frac{1}{2 ^{k}}) - 4 k + 4 ln (2) k^{2} - 8 k^{2}

Apply log rule:

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

ln (\frac{1}{2 ^{k}}) = ln (1) - ln (2^{k})

= 4 (ln (1) - ln (2^{k})) - 4 k + 4 ln (2) k^{2} - 8 k^{2}

Apply log rule:

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

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

= 4 (ln (1) - k ln (2)) - 4 k + 4 ln (2) k^{2} - 8 k^{2}

4 (ln (1) - k ln (2)) = - 4 ln (2) k

= - 4 ln (2) k - 4 k + 4 ln (2) k^{2} - 8 k^{2}

\frac{1}{200} ln (80)

ln (\frac{760}{600})

- lo g_{10} (1.833 \cdot 1 0^{- 5})

ln (\frac{27}{40})

\frac{3}{2} lo g_{2} (5)

What is 4ln(1/(2^k))-4k+4ln(2)k^2-8k^2 ?
The solution to 4ln(1/(2^k))-4k+4ln(2)k^2-8k^2 is -4ln(2)k-4k+4ln(2)k^2-8k^2