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Popular >

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

Solution

simplify

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

Solution

- 4 ln (2) - ln (3)

+1

Decimal Notation

- 3.87120 \dots

Solution steps

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

Apply log rule:

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

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

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

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 \cdot 4^{3})

Apply log rule:

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

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

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

Apply log rule:

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

ln (4^{3}) = 3 ln (4)

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

3 ln (4) = 6 ln (2)

= 2 ln (2) - (6 ln (2) + ln (3))

- (ln (3) + 6 ln (2)) : - ln (3) - 6 ln (2)

= 2 ln (2) - ln (3) - 6 ln (2)

Simplify

2 ln (2) - ln (3) - 6 ln (2) : - 4 ln (2) - ln (3)

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

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Frequently Asked Questions (FAQ)

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