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

The solution to 2ln(3)-ln(45)+2ln(5) is ln(5)

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

simplify

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

ln (5)

Decimal Notation

1.60943 \dots

Solution steps

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

Apply log rule:

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

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

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

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

\frac{3 ^{2}}{45} = \frac{1}{5}

= ln (\frac{1}{5}) + 2 ln (5)

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

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

Add similar elements:

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

= ln (5)

ln (\frac{497}{4888})

ln (\frac{373}{293})

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ln (45) + ln (19)

ln (w^{3}) - ln (z)