What is 2ln(3)-2ln(5/3) ?

The solution to 2ln(3)-2ln(5/3) is ln(81/25)

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simplify 2ln(3)-2ln(5/3)

simplify

2 ln (3) - 2 ln (\frac{5}{3})

ln (\frac{81}{25})

Decimal Notation

1.17557 \dots

Solution steps

2 ln (3) - 2 ln (\frac{5}{3})

Apply log rule:

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

2 ln (3) = ln (3^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

\frac{3 ^{2}}{( \frac{5}{3} ) ^{2}} = \frac{81}{25}

= ln (\frac{81}{25})

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

ln (\frac{1.6}{0.277})

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