What is expand ln(1/(9k)) ?

The solution to expand ln(1/(9k)) is -ln(k)-2ln(3)

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expand ln(1/(9k))

expand

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

- ln (k) - 2 ln (3)

Solution steps

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

Apply log rule:

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

ln (\frac{1}{9 k}) = ln (1) - ln (9 k)

= ln (1) - ln (9 k)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln (9 k) = ln (9) + ln (k)

= ln (1) - (ln (k) + ln (9))

Apply log rule:

lo g_{a} (1) = 0

= 0 - (ln (k) + ln (9))

ln (9) = 2 ln (3)

= 0 - (ln (k) + 2 ln (3))

0 - (2 ln (3) + ln (k)) = - (2 ln (3) + ln (k))

= - (ln (k) + 2 ln (3))

Distribute parentheses

= - (ln (k)) - (2 ln (3))

Apply minus-plus rules

+ (- a) = - a

= - ln (k) - 2 ln (3)

lo g_{b} (z^{5} x)

lo g_{10} (x) - 8 lo g_{10} (y) + 9 lo g_{10} (z)

lo g_{10} (7 a) + lo g_{10} (6 b)

3 (lo g_{7} (x) + 3 lo g_{7} (y) - 4 lo g_{7} (z))

2 ln (x) + 4 ln (y) - 5 ln (z)