What is 1/3 ln|3*2-2|-1/3 ln|5| ?

The solution to 1/3 ln|3*2-2|-1/3 ln|5| is 1/3 ln(4/5)

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simplify 1/3 ln|3*2-2|-1/3 ln|5|

simplify

\frac{1}{3} ln ∣ 3 \cdot 2 - 2 ∣ - \frac{1}{3} ln ∣ 5 ∣

\frac{1}{3} ln (\frac{4}{5})

Decimal Notation

- 0.07438 \dots

Solution steps

\frac{1}{3} ln ∣ 3 \cdot 2 - 2 ∣ - \frac{1}{3} ln ∣ 5 ∣

Apply log rule:

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

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

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

Apply log rule:

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

\frac{1}{3} ln ∣ 5 ∣ = ln (∣ 5 ∣^{\frac{1}{3}})

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

Apply log rule:

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

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

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

Rewrite as

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

Apply log rule

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

assuming

x \geq 0

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

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What is 1/3 ln|3*2-2|-1/3 ln|5| ?
The solution to 1/3 ln|3*2-2|-1/3 ln|5| is 1/3 ln(4/5)