What is expand ln(1/(4^k)) ?

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

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expand ln(1/(4^k))

expand

ln (\frac{1}{4 ^{k}})

- 2 ln (2) k

Solution steps

ln (\frac{1}{4 ^{k}})

Apply log rule:

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

ln (\frac{1}{4 ^{k}}) = ln (1) - ln (4^{k})

= ln (1) - ln (4^{k})

Apply log rule:

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

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

= ln (1) - k ln (4)

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (4) k

0 - k ln (4) = - k ln (4)

= - ln (4) k

Simplify

ln (4) : 2 ln (2)

= - 2 ln (2) k

lo g_{9} (9 x)

lo g_{10} (x) - \frac{1}{2} lo g_{10} (y) + 5 lo g_{10} (z)

ln (5) + 2 ln (x) + 3 ln (x^{2} + 5)

ln (\frac{1}{9 ^{k}})

lo g_{3} (9 x y^{2}) - lo g_{3} (27 x y)