What is 1/5 ln(2)+4/5 ln(3/4) ?

The solution to 1/5 ln(2)+4/5 ln(3/4) is 1/5 ln(81/128)

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simplify 1/5 ln(2)+4/5 ln(3/4)

simplify

\frac{1}{5} ln (2) + \frac{4}{5} ln (\frac{3}{4})

\frac{1}{5} ln (\frac{81}{128})

Decimal Notation

- 0.09151 \dots

Solution steps

\frac{1}{5} ln (2) + \frac{4}{5} ln (\frac{3}{4})

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply exponent rule:

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

Rewrite as

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

(\frac{3}{4})^{4} = \frac{3 ^{4}}{4 ^{4}}

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

Multiply

2 \cdot \frac{3 ^{4}}{4 ^{4}} : \frac{81}{128}

= \frac{1}{5} ln (\frac{81}{128})

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What is 1/5 ln(2)+4/5 ln(3/4) ?
The solution to 1/5 ln(2)+4/5 ln(3/4) is 1/5 ln(81/128)