What is 3/5 ln(3)+2/5 ln(2) ?

The solution to 3/5 ln(3)+2/5 ln(2) is 1/5 ln(108)

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simplify 3/5 ln(3)+2/5 ln(2)

simplify

\frac{3}{5} ln (3) + \frac{2}{5} ln (2)

\frac{1}{5} ln (108)

Decimal Notation

0.93642 \dots

Solution steps

\frac{3}{5} ln (3) + \frac{2}{5} ln (2)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply exponent rule:

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

Rewrite as

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

3^{3} = 27

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

2^{2} = 4

= \frac{1}{5} ln (27 \cdot 4)

Multiply the numbers:

27 \cdot 4 = 108

= \frac{1}{5} ln (108)

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What is 3/5 ln(3)+2/5 ln(2) ?
The solution to 3/5 ln(3)+2/5 ln(2) is 1/5 ln(108)