What is ln((3a^3b^2)/(sqrt(a))) ?

The solution to ln((3a^3b^2)/(sqrt(a))) is ln(3)+3ln(a)+2ln(b)-ln(sqrt(a))

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simplify ln((3a^3b^2)/(sqrt(a)))

simplify

ln (\frac{3 a ^{3} b ^{2}}{a})

ln (3) + 3 ln (a) + 2 ln (b) - ln (a)

Solution steps

ln (\frac{3 a ^{3} b ^{2}}{a})

Apply log rule:

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

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

= ln (3 a^{3} b^{2}) - ln (a)

Apply log rule:

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

ln (3 a^{3} b^{2}) = ln (3) + ln (a^{3}) + ln (b^{2})

= ln (3) + ln (a^{3}) + ln (b^{2}) - ln (a)

Apply log rule:

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

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

= ln (3) + 3 ln (a) + ln (b^{2}) - ln (a)

Apply log rule:

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

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

= ln (3) + 3 ln (a) + 2 ln (b) - ln (a)

lo g_{6} (10) + lo g_{6} (15)

ln (\frac{6}{e ^{2}})

2 lo g_{10} (x) + 3 lo g_{10} (x + 1)

ln (\frac{1 0 ^{- 4} x ^{- 2}}{x ^{5}})

- \frac{2}{5} lo g_{2} (\frac{2}{5}) - \frac{3}{5} lo g_{2} (\frac{3}{5})

What is ln((3a^3b^2)/(sqrt(a))) ?
The solution to ln((3a^3b^2)/(sqrt(a))) is ln(3)+3ln(a)+2ln(b)-ln(sqrt(a))