What is ln(2)ln(3)-ln(45)+2ln(5) ?

The solution to ln(2)ln(3)-ln(45)+2ln(5) is ln(5/(3^{-ln(2)+2)})

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simplify ln(2)ln(3)-ln(45)+2ln(5)

simplify

ln (2) ln (3) - ln (45) + 2 ln (5)

ln (\frac{5}{3 ^{- l n (2) + 2}})

Solution steps

ln (2) ln (3) - ln (45) + 2 ln (5)

Apply log rule:

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

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

= ln (3^{l n (2)}) - ln (45) + 2 ln (5)

Apply log rule:

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

ln (3^{l n (2)}) - ln (45) = ln (\frac{3 ^{l n (2)}}{45})

= ln (\frac{3 ^{l n (2)}}{45}) + 2 ln (5)

\frac{3 ^{l n (2)}}{45} = \frac{1}{5 \cdot 3 ^{- l n (2) + 2}}

= ln (\frac{1}{5 \cdot 3 ^{- l n (2) + 2}}) + 2 ln (5)

Apply log rule:

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

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

= ln (\frac{1}{5 \cdot 3 ^{- l n (2) + 2}}) + ln (5^{2})

Apply log rule:

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

ln (\frac{1}{5 \cdot 3 ^{- l n (2) + 2}}) + ln (5^{2}) = ln (5^{2} \cdot \frac{1}{5 \cdot 3 ^{- l n (2) + 2}})

= ln (5^{2} \cdot \frac{1}{5 \cdot 3 ^{- l n (2) + 2}})

\frac{1}{5 \cdot 3 ^{- l n (2) + 2}} \cdot 5^{2} = \frac{5}{3 ^{- l n (2) + 2}}

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

ln (\frac{2 ^{- 4}}{3 ^{1} 5 ^{4}})

- ln (\frac{- x}{A \cdot k})

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What is ln(2)ln(3)-ln(45)+2ln(5) ?
The solution to ln(2)ln(3)-ln(45)+2ln(5) is ln(5/(3^{-ln(2)+2)})