What is 2ln(a)-1/3 ln(b)+3ln(c) ?

The solution to 2ln(a)-1/3 ln(b)+3ln(c) is ln((a^2c^3b^{2/3})/b)

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simplify 2ln(a)-1/3 ln(b)+3ln(c)

simplify

2 ln (a) - \frac{1}{3} ln (b) + 3 ln (c)

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

Solution steps

2 ln (a) - \frac{1}{3} ln (b) + 3 ln (c)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

= ln (a^{2}) - \frac{1}{3} ln (b) + 3 ln (c)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} ln (b) = ln (b^{\frac{1}{3}})

= ln (a^{2}) - ln (b^{\frac{1}{3}}) + 3 ln (c)

Apply log rule:

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

ln (a^{2}) - ln (b^{\frac{1}{3}}) = ln (\frac{a ^{2}}{b ^{\frac{1}{3}}})

= ln (\frac{a ^{2}}{b ^{\frac{1}{3}}}) + 3 ln (c)

Simplify

\frac{a ^{2}}{b ^{\frac{1}{3}}} : \frac{a ^{2} b ^{\frac{2}{3}}}{b}

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

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

Apply log rule:

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

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

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

Simplify

\frac{a ^{2} b ^{\frac{2}{3}}}{b} c^{3} : \frac{a ^{2} c ^{3} b ^{\frac{2}{3}}}{b}

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

lo g_{10} (\frac{4 c}{3})

lo g_{10} (\frac{1000}{m ^{2} + n ^{2}})

lo g_{4} (3 x y)

5 lo g_{5} (x) - \frac{1}{4} lo g_{5} (8 - x)

lo g_{10} (d) - 4 lo g_{10} (g)

What is 2ln(a)-1/3 ln(b)+3ln(c) ?
The solution to 2ln(a)-1/3 ln(b)+3ln(c) is ln((a^2c^3b^{2/3})/b)