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

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

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

simplify

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

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

Decimal Notation

- 3.74212 \dots

Solution steps

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

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

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

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^{2})

Apply log rule:

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

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

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

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^{2}))

Apply log rule:

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

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

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

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

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

lo g_{a} (x \times y)

lo g_{2} (a) - lo g_{2} (b)

lo g_{b} (5 x \cdot 3 y)

lo g_{b} (3) + lo g_{b} (6)

lo g_{10} (583) - \frac{1}{2} lo g_{10} (150)

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