What is ln((2^2)/(3^45^3)) ?

The solution to ln((2^2)/(3^45^3)) is 2ln(2)-4ln(3)-3ln(5)

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simplify ln((2^2)/(3^45^3))

simplify

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

2 ln (2) - 4 ln (3) - 3 ln (5)

Decimal Notation

- 7.83646 \dots

Solution steps

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

Apply log rule:

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

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

= ln (2^{2}) - ln (3^{4} \cdot 5^{3})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= 2 ln (2) - ln (3^{4} \cdot 5^{3})

Apply log rule:

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

ln (3^{4} \cdot 5^{3}) = ln (3^{4}) + ln (5^{3})

= 2 ln (2) - (ln (3^{4}) + ln (5^{3}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (3^{4}) = 4 ln (3)

= 2 ln (2) - (4 ln (3) + ln (5^{3}))

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (5^{3}) = 3 ln (5)

= 2 ln (2) - (4 ln (3) + 3 ln (5))

- (4 ln (3) + 3 ln (5)) : - 4 ln (3) - 3 ln (5)

= 2 ln (2) - 4 ln (3) - 3 ln (5)

2 ln (x) + ln (z)

- lo g_{10} (2.95 \cdot 1 0^{- 2})

ln (cot (x) \cdot e^{x})

ln (x - 3) + ln (x - 6)

ln (\frac{6}{60})

What is ln((2^2)/(3^45^3)) ?
The solution to ln((2^2)/(3^45^3)) is 2ln(2)-4ln(3)-3ln(5)