What is 3/8*ln(8/3)+5/16*ln(16/5)+5/16*ln(16/5) ?

The solution to 3/8*ln(8/3)+5/16*ln(16/5)+5/16*ln(16/5) is 1/8 ln(536870912/84375)

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simplify 3/8ln(8/3)+5/16ln(16/5)+5/16*ln(16/5)

simplify

\frac{3}{8} \cdot ln (\frac{8}{3}) + \frac{5}{16} \cdot ln (\frac{16}{5}) + \frac{5}{16} \cdot ln (\frac{16}{5})

\frac{1}{8} ln (\frac{536870912}{84375})

Decimal Notation

1.09478 \dots

Solution steps

\frac{3}{8} ln (\frac{8}{3}) + \frac{5}{16} ln (\frac{16}{5}) + \frac{5}{16} ln (\frac{16}{5})

Apply log rule:

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

\frac{3}{8} ln (\frac{8}{3}) = ln ((\frac{8}{3})^{\frac{3}{8}})

= ln ((\frac{8}{3})^{\frac{3}{8}}) + \frac{5}{16} ln (\frac{16}{5}) + \frac{5}{16} ln (\frac{16}{5})

Apply log rule:

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

\frac{5}{16} ln (\frac{16}{5}) = ln ((\frac{16}{5})^{\frac{5}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}}) + ln ((\frac{16}{5})^{\frac{5}{16}}) + \frac{5}{16} ln (\frac{16}{5})

Apply log rule:

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

ln ((\frac{8}{3})^{\frac{3}{8}}) + ln ((\frac{16}{5})^{\frac{5}{16}}) = ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}}) + \frac{5}{16} ln (\frac{16}{5})

Apply log rule:

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

\frac{5}{16} ln (\frac{16}{5}) = ln ((\frac{16}{5})^{\frac{5}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}}) + ln ((\frac{16}{5})^{\frac{5}{16}})

Apply log rule:

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

ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}}) + ln ((\frac{16}{5})^{\frac{5}{16}}) = ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}} (\frac{16}{5})^{\frac{5}{16}})

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}} (\frac{16}{5})^{\frac{5}{16}})

(\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{16}} (\frac{16}{5})^{\frac{5}{16}} = (\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{8}}

= ln ((\frac{8}{3})^{\frac{3}{8}} (\frac{16}{5})^{\frac{5}{8}})

Apply exponent rule:

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

Rewrite as

= ln ((\frac{8}{3})^{3} (\frac{16}{5})^{5})^{\frac{1}{8}}

Apply log rule

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

assuming

x \geq 0

= \frac{1}{8} ln ((\frac{8}{3})^{3} (\frac{16}{5})^{5})

(\frac{8}{3})^{3} = \frac{8 ^{3}}{3 ^{3}}

= \frac{1}{8} ln ((\frac{16}{5})^{5} \frac{8 ^{3}}{3 ^{3}})

(\frac{16}{5})^{5} = \frac{1 6 ^{5}}{5 ^{5}}

= \frac{1}{8} ln (\frac{8 ^{3}}{3 ^{3}} \cdot \frac{1 6 ^{5}}{5 ^{5}})

Multiply

\frac{8 ^{3}}{3 ^{3}} \cdot \frac{1 6 ^{5}}{5 ^{5}} : \frac{536870912}{84375}

= \frac{1}{8} ln (\frac{536870912}{84375})

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

ln (\frac{3 ^{x}}{( 1 + x ^{3} ) ^{2}})

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What is 3/8*ln(8/3)+5/16*ln(16/5)+5/16*ln(16/5) ?
The solution to 3/8*ln(8/3)+5/16*ln(16/5)+5/16*ln(16/5) is 1/8 ln(536870912/84375)