What is ln((2^4)/(3^45^3)) ?

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

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simplify ln((2^4)/(3^45^3))

simplify

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

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

Decimal Notation

- 6.45017 \dots

Solution steps

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

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

= 4 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)

= 4 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)

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

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

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

ln (4 π)

lo g_{3} (c) + lo g_{3} (c + 2)

\frac{1}{3} lo g_{2} (x) + \frac{1}{2} lo g_{2} (y) + lo g_{2} (z)

ln (2) + ln (9)

3 ln (x + 4) + 9 ln (x - 4) - 5 ln (x)

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