What is 3/5 ln|12|-2/5 ln|6| ?

The solution to 3/5 ln|12|-2/5 ln|6| is 1/5 ln(48)

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simplify 3/5 ln|12|-2/5 ln|6|

simplify

\frac{3}{5} ln ∣ 12 ∣ - \frac{2}{5} ln ∣ 6 ∣

\frac{1}{5} ln (48)

Decimal Notation

0.77424 \dots

Solution steps

\frac{3}{5} ln ∣ 12 ∣ - \frac{2}{5} ln ∣ 6 ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{3}{5} ln ∣ 12 ∣ = ln (∣ 12 ∣^{\frac{3}{5}})

= ln (∣ 12 ∣^{\frac{3}{5}}) - \frac{2}{5} ln ∣ 6 ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{2}{5} ln ∣ 6 ∣ = ln (∣ 6 ∣^{\frac{2}{5}})

= ln (∣ 12 ∣^{\frac{3}{5}}) - ln (∣ 6 ∣^{\frac{2}{5}})

Apply log rule:

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

ln (∣ 12 ∣^{\frac{3}{5}}) - ln (∣ 6 ∣^{\frac{2}{5}}) = ln (\frac{∣ 12 ∣ ^{\frac{3}{5}}}{∣ 6 ∣ ^{\frac{2}{5}}})

= ln (\frac{∣ 12 ∣ ^{\frac{3}{5}}}{∣ 6 ∣ ^{\frac{2}{5}}})

\frac{∣ 12 ∣ ^{\frac{3}{5}}}{∣ 6 ∣ ^{\frac{2}{5}}} = 2^{\frac{4}{5}} \cdot 3^{\frac{1}{5}}

= ln (2^{\frac{4}{5}} \cdot 3^{\frac{1}{5}})

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

Rewrite as

= ln ((2^{4} \cdot 3)^{\frac{1}{5}})

Apply log rule

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

assuming

x \geq 0

= \frac{1}{5} ln (2^{4} \cdot 3)

2^{4} = 16

= \frac{1}{5} ln (16 \cdot 3)

Multiply the numbers:

16 \cdot 3 = 48

= \frac{1}{5} ln (48)

ln ((x^{2} + 1) (x^{3} + 2)^{6})

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lo g_{4} (1 e^{20})

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

\frac{7}{30} (ln (7) - ln (2))

What is 3/5 ln|12|-2/5 ln|6| ?
The solution to 3/5 ln|12|-2/5 ln|6| is 1/5 ln(48)