What is ln(3/4)*3/4+ln(1/4)*1/4 ?

The solution to ln(3/4)*3/4+ln(1/4)*1/4 is 1/4 ln(27/256)

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

simplify

ln (\frac{3}{4}) \cdot \frac{3}{4} + ln (\frac{1}{4}) \cdot \frac{1}{4}

\frac{1}{4} ln (\frac{27}{256})

Decimal Notation

- 0.56233 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply exponent rule:

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

Rewrite as

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

Apply log rule

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

assuming

x \geq 0

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

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

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

Multiply

\frac{3 ^{3}}{4 ^{3}} \cdot \frac{1}{4} : \frac{27}{256}

= \frac{1}{4} ln (\frac{27}{256})

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- lo g_{10} (7.98 \cdot 1 0^{- 2})

ln (e^{x}) + ln (2)

9 \cdot ln (- 1155 \cdot 3)

- lo g_{10} (5.9 \cdot 1 0^{- 7})

What is ln(3/4)*3/4+ln(1/4)*1/4 ?
The solution to ln(3/4)*3/4+ln(1/4)*1/4 is 1/4 ln(27/256)