What is ln(1/(2t^3)) ?

The solution to ln(1/(2t^3)) is -3ln(t)-ln(2)

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simplify ln(1/(2t^3))

simplify

ln (\frac{1}{2 t ^{3}})

- 3 ln (t) - ln (2)

Solution steps

ln (\frac{1}{2 t ^{3}})

Apply log rule:

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

ln (\frac{1}{2 t ^{3}}) = ln (1) - ln (2 t^{3})

= ln (1) - ln (2 t^{3})

Apply log rule:

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

ln (2 t^{3}) = ln (2) + ln (t^{3})

= ln (1) - (ln (t^{3}) + ln (2))

Apply log rule:

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

ln (t^{3}) = 3 ln (t)

= ln (1) - (ln (2) + 3 ln (t))

Apply log rule:

lo g_{a} (1) = 0

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

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

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

Distribute parentheses

= - (3 ln (t)) - (ln (2))

Apply minus-plus rules

+ (- a) = - a

= - 3 ln (t) - ln (2)

- ln (\frac{1000 - x}{9 x})

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ln (\frac{x ^{2} + 1}{x + 2})

ln (3 a)