simplify ln(1/(9k))

Solution

simplify

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)

3 (2 \cdot lo g_{10} (a) - lo g_{10} (b))

lo g_{8} (\frac{x y ^{5}}{z ^{6}})

6 ln (x) - \frac{1}{8} ln (y)

- lo g_{10} (4 \times 1 0^{- 5})

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