What is 3/5 ln|4-1|+2/5 ln|4+4| ?

The solution to 3/5 ln|4-1|+2/5 ln|4+4| is 3/5 ln(12)

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simplify 3/5 ln|4-1|+2/5 ln|4+4|

simplify

\frac{3}{5} ln ∣ 4 - 1 ∣ + \frac{2}{5} ln ∣ 4 + 4 ∣

\frac{3}{5} ln (12)

Decimal Notation

1.49094 \dots

Solution steps

\frac{3}{5} ln ∣ 4 - 1 ∣ + \frac{2}{5} ln ∣ 4 + 4 ∣

Apply log rule:

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

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

= ln (∣ 4 - 1 ∣^{\frac{3}{5}}) + \frac{2}{5} ln ∣ 4 + 4 ∣

Apply log rule:

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

\frac{2}{5} ln ∣ 4 + 4 ∣ = ln (∣ 4 + 4 ∣^{\frac{2}{5}})

= ln (∣ 4 - 1 ∣^{\frac{3}{5}}) + ln (∣ 4 + 4 ∣^{\frac{2}{5}})

Apply log rule:

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

ln (∣ 4 - 1 ∣^{\frac{3}{5}}) + ln (∣ 4 + 4 ∣^{\frac{2}{5}}) = ln (∣ 4 - 1 ∣^{\frac{3}{5}} ∣ 4 + 4 ∣^{\frac{2}{5}})

= ln (∣ 4 - 1 ∣^{\frac{3}{5}} ∣ 4 + 4 ∣^{\frac{2}{5}})

∣ 4 - 1 ∣^{\frac{3}{5}} ∣ 4 + 4 ∣^{\frac{2}{5}} = 3^{\frac{3}{5}} \cdot 8^{\frac{2}{5}}

= ln (3^{\frac{3}{5}} \cdot 8^{\frac{2}{5}})

Apply exponent rule:

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

Rewrite as

= ln ((3^{3} \cdot 8^{2})^{\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 (3^{3} \cdot 8^{2})

3^{3} = 27

= \frac{1}{5} ln (8^{2} \cdot 27)

8^{2} = 64

= \frac{1}{5} ln (27 \cdot 64)

Multiply the numbers:

27 \cdot 64 = 1728

= \frac{1}{5} ln (1728)

Simplify

ln (1728) : 3 ln (12)

= 3 \cdot \frac{1}{5} ln (12)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 3}{5} ln (12)

Multiply the numbers:

1 \cdot 3 = 3

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

4 ln (1) - 3 ln (2) - 4 ln (\frac{1}{2}) - 3 ln (\frac{1}{2} + 1)

ln (\frac{25}{36})

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

lo g_{8} (\frac{1}{625})

ln (\frac{9.96087}{0.46087})

What is 3/5 ln|4-1|+2/5 ln|4+4| ?
The solution to 3/5 ln|4-1|+2/5 ln|4+4| is 3/5 ln(12)