What is 5+2ln(2)-2ln(5) ?

The solution to 5+2ln(2)-2ln(5) is 5+ln(4/25)

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simplify 5+2ln(2)-2ln(5)

simplify

5 + 2 ln (2) - 2 ln (5)

5 + ln (\frac{4}{25})

Decimal Notation

3.16741 \dots

Solution steps

5 + 2 ln (2) - 2 ln (5)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

\frac{2 ^{2}}{5 ^{2}} = \frac{4}{25}

= 5 + ln (\frac{4}{25})

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