What is ln(30)+2ln(5)-ln(15) ?

The solution to ln(30)+2ln(5)-ln(15) is ln(50)

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simplify ln(30)+2ln(5)-ln(15)

simplify

ln (30) + 2 ln (5) - ln (15)

ln (50)

Decimal Notation

3.91202 \dots

Solution steps

ln (30) + 2 ln (5) - ln (15)

Apply log rule:

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

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

= ln (30) + ln (5^{2}) - ln (15)

Apply log rule:

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

ln (30) + ln (5^{2}) = ln (5^{2} \cdot 30)

= ln (5^{2} \cdot 30) - ln (15)

30 \cdot 5^{2} = 750

= ln (750) - ln (15)

Apply log rule:

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

ln (750) - ln (15) = ln (\frac{750}{15})

= ln (\frac{750}{15})

Divide the numbers:

\frac{750}{15} = 50

= ln (50)

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