What is ln((2^4)/(3^35^3)) ?

The solution to ln((2^4)/(3^35^3)) is 4ln(2)-3ln(3)-3ln(5)

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simplify ln((2^4)/(3^35^3))

simplify

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

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

Decimal Notation

- 5.35156 \dots

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

ln (2^{4}) = 4 ln (2)

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= 4 ln (2) - (3 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)

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

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

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

lo g_{10} (5. 1^{2}) + lo g_{10} (2. 6^{4})

3 ln (\frac{5}{2}) + 4 ln (\frac{14}{11})

ln (\frac{18}{22})

ln (5 e) - ln (1 0^{3}) + lo g_{2} (4^{2})

3 ln (x - 8) - 4 ln (x)

What is ln((2^4)/(3^35^3)) ?
The solution to ln((2^4)/(3^35^3)) is 4ln(2)-3ln(3)-3ln(5)