What is 3/16 [1ln(1)+4ln(4/3)+5ln(5/3)+2ln(2)] ?

The solution to 3/16 [1ln(1)+4ln(4/3)+5ln(5/3)+2ln(2)] is 3/16 ln(3200000/19683)

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simplify 3/16 [1ln(1)+4ln(4/3)+5ln(5/3)+2ln(2)]

simplify

\frac{3}{16} [1 ln (1) + 4 ln (\frac{4}{3}) + 5 ln (\frac{5}{3}) + 2 ln (2)]

\frac{3}{16} ln (\frac{3200000}{19683})

Decimal

0.95459 \dots

Solution steps

\frac{3}{16} [1 \cdot ln (1) + 4 ln (\frac{4}{3}) + 5 ln (\frac{5}{3}) + 2 ln (2)]

Apply log rule:

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

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

= \frac{3}{16} [1 \cdot ln (1) + 5 ln (\frac{5}{3}) + 2 ln (2) + ln ((\frac{4}{3})^{4})]

Apply log rule:

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

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

= \frac{3}{16} [1 \cdot ln (1) + 2 ln (2) + ln ((\frac{4}{3})^{4}) + ln ((\frac{5}{3})^{5})]

Apply log rule:

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

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

= \frac{3}{16} [1 \cdot ln (1) + 2 ln (2) + ln ((\frac{5}{3})^{5} (\frac{4}{3})^{4})]

(\frac{4}{3})^{4} (\frac{5}{3})^{5} = \frac{800000}{19683}

= \frac{3}{16} [1 \cdot ln (1) + ln (\frac{800000}{19683}) + 2 ln (2)]

Apply log rule:

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

2 ln (2) = ln (2^{2})

= \frac{3}{16} [1 \cdot ln (1) + ln (2^{2}) + ln (\frac{800000}{19683})]

Apply log rule:

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

ln (\frac{800000}{19683}) + ln (2^{2}) = ln (2^{2} \cdot \frac{800000}{19683})

= \frac{3}{16} [1 \cdot ln (1) + ln (2^{2} \cdot \frac{800000}{19683})]

\frac{800000}{19683} \cdot 2^{2} = \frac{3200000}{19683}

= \frac{3}{16} [1 \cdot ln (1) + ln (\frac{3200000}{19683})]

1 \cdot ln (1) = 0

= \frac{3}{16} (ln (\frac{3200000}{19683}) + 0)

0 + ln (\frac{3200000}{19683}) = ln (\frac{3200000}{19683})

= \frac{3}{16} ln (\frac{3200000}{19683})

ln (\frac{1980 - 176}{894 - 176}) \cdot \frac{1}{3.058}

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\frac{ln ( 3 ) + ln ( h ) - ln ( 3 )}{h}

ln (x) + \frac{1}{2} ln (1 + u)

5 lo g_{5} (5 z + 1) + \frac{1}{2} lo g_{5} (z + 8)

What is 3/16 [1ln(1)+4ln(4/3)+5ln(5/3)+2ln(2)] ?
The solution to 3/16 [1ln(1)+4ln(4/3)+5ln(5/3)+2ln(2)] is 3/16 ln(3200000/19683)