What is 5ln(2)-2ln(7/2) ?

The solution to 5ln(2)-2ln(7/2) is ln(128/49)

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simplify 5ln(2)-2ln(7/2)

simplify

5 ln (2) - 2 ln (\frac{7}{2})

ln (\frac{128}{49})

Decimal Notation

0.96020 \dots

Solution steps

5 ln (2) - 2 ln (\frac{7}{2})

Apply log rule:

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

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

= ln (2^{5}) - 2 ln (\frac{7}{2})

Apply log rule:

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

2 ln (\frac{7}{2}) = ln ((\frac{7}{2})^{2})

= ln (2^{5}) - ln ((\frac{7}{2})^{2})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (2^{5}) - ln ((\frac{7}{2})^{2}) = ln (\frac{2 ^{5}}{( \frac{7}{2} ) ^{2}})

= ln (\frac{2 ^{5}}{( \frac{7}{2} ) ^{2}})

\frac{2 ^{5}}{( \frac{7}{2} ) ^{2}} = \frac{128}{49}

= ln (\frac{128}{49})

ln (2022) - ln (2020)

2 ln (a) + 5 ln (b) - 7 ln (c)

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

ln (\frac{115}{100})

- lo g_{10} (2.4 \cdot 1 0^{- 6})