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

The solution to ln((3a^3a^2)/(sqrt(a))) is ln(3)+5ln(a)-ln(sqrt(a))

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

simplify

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

ln (3) + 5 ln (a) - ln (a)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Add similar elements:

3 ln (a) + 2 ln (a) = 5 ln (a)

= ln (3) + 5 ln (a) - ln (a)

lo g_{10} (\frac{x}{100 y ^{6}})

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

lo g_{2} (12) + lo g_{2} (3) - 2 lo g_{2} (2)

lo g_{10} (10 x^{5})

lo g_{5} (25 z)

What is ln((3a^3a^2)/(sqrt(a))) ?
The solution to ln((3a^3a^2)/(sqrt(a))) is ln(3)+5ln(a)-ln(sqrt(a))