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

The solution to ln(5)+ln(3)-2ln(2) is ln(15/4)

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

simplify

ln (5) + ln (3) - 2 ln (2)

ln (\frac{15}{4})

Decimal Notation

1.32175 \dots

Solution steps

ln (5) + ln (3) - 2 ln (2)

Apply log rule:

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

ln (5) + ln (3) = ln (5 \cdot 3)

= ln (5 \cdot 3) - 2 ln (2)

Multiply the numbers:

5 \cdot 3 = 15

= ln (15) - 2 ln (2)

Apply log rule:

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

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

= ln (15) - ln (2^{2})

Apply log rule:

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

ln (15) - ln (2^{2}) = ln (\frac{15}{2 ^{2}})

= ln (\frac{15}{2 ^{2}})

2^{2} = 4

= ln (\frac{15}{4})

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