What is ln(((a^4))/(b^{-2)c^2}) ?

The solution to ln(((a^4))/(b^{-2)c^2}) is 4ln(a)+2ln(b)-2ln(c)

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simplify ln(((a^4))/(b^{-2)c^2})

simplify

ln (\frac{( a ^{4} )}{b ^{- 2} c ^{2}})

4 ln (a) + 2 ln (b) - 2 ln (c)

Solution steps

ln (\frac{( a ^{4} )}{b ^{- 2} c ^{2}})

Apply log rule:

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

ln (\frac{( a ^{4} )}{b ^{- 2} c ^{2}}) = ln ((a^{4})) - ln (b^{- 2} c^{2})

= ln (a^{4}) - ln (b^{- 2} c^{2})

Apply log rule:

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

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

= 4 ln (a) - ln (b^{- 2} c^{2})

Apply log rule:

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

ln (b^{- 2} c^{2}) = ln (b^{- 2}) + ln (c^{2})

= 4 ln (a) - (ln (b^{- 2}) + ln (c^{2}))

Apply log rule:

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

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

= 4 ln (a) - (- 2 ln (b) + ln (c^{2}))

Apply log rule:

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

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

= 4 ln (a) - (- 2 ln (b) + 2 ln (c))

- (- 2 ln (b) + 2 ln (c)) : 2 ln (b) - 2 ln (c)

= 4 ln (a) + 2 ln (b) - 2 ln (c)

10 lo g_{10} (\frac{1 0 ^{12} \cdot 0.5}{4 \cdot π \cdot 20 0 ^{2}})

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3727.8 \cdot ln (\frac{( 23180 + 56088 )}{23180})

\frac{1}{3} ln (x) + \frac{1}{3} ln (3 x)

2 ln (3) - 4 ln (7)

What is ln(((a^4))/(b^{-2)c^2}) ?
The solution to ln(((a^4))/(b^{-2)c^2}) is 4ln(a)+2ln(b)-2ln(c)