What is ln(((2^{(-2)}))/((3^{(2))5^{(4)})}) ?

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

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simplify ln(((2^{(-2)}))/((3^{(2))5^{(4)})})

simplify

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

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

Decimal Notation

- 10.02127 \dots

Solution steps

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

Apply log rule:

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

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

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

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^{2} \cdot 5^{4})

Apply log rule:

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

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

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

Apply log rule:

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

ln (3^{2}) = 2 ln (3)

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

Apply log rule:

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

ln (5^{4}) = 4 ln (5)

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

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

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

\frac{1}{4} ln (2 + e^{3 x}) - \frac{5}{12} ln (2) - 15 \frac{x}{12} + c

- lo g_{10} (0.1 M)

x + 4 lo g_{5} (y) - 2 lo g_{5} (z)

- \cdot ln (\frac{v _{1}}{v _{4}})

ln (\frac{643}{30})

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