What is ln((12x)/(4x)) ?

The solution to ln((12x)/(4x)) is -2ln(2)+ln(12)

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simplify ln((12x)/(4x))

simplify

ln (\frac{12 x}{4 x})

- 2 ln (2) + ln (12)

Decimal Notation

1.09861 \dots

Solution steps

ln (\frac{12 x}{4 x})

Apply log rule:

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

ln (\frac{12 x}{4 x}) = ln (12 x) - ln (4 x)

= ln (12 x) - ln (4 x)

Apply log rule:

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

ln (12 x) = ln (12) + ln (x), ln (4 x) = ln (4) + ln (x)

= ln (12) + ln (x) - (ln (x) + ln (4))

ln (4) = 2 ln (2)

= ln (12) + ln (x) - (ln (x) + 2 ln (2))

- (2 ln (2) + ln (x)) : - 2 ln (2) - ln (x)

= ln (12) + ln (x) - 2 ln (2) - ln (x)

Simplify

ln (12) + ln (x) - 2 ln (2) - ln (x) : - 2 ln (2) + ln (12)

= - 2 ln (2) + ln (12)

ln (\frac{1}{\frac{x}{y}})

ln (y) + ln (1 - x^{2})

\frac{1}{2 - 4} ln (\frac{2}{4})

5 lo g_{3} (x) + 3 lo g_{3} (y)

ln (\frac{40}{11})