What is ln(a)-3ln(b)+6ln(c) ?

The solution to ln(a)-3ln(b)+6ln(c) is ln((ac^6)/(b^3))

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simplify ln(a)-3ln(b)+6ln(c)

simplify

ln (a) - 3 ln (b) + 6 ln (c)

ln (\frac{a c ^{6}}{b ^{3}})

Solution steps

ln (a) - 3 ln (b) + 6 ln (c)

Apply log rule:

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

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

= ln (a) - ln (b^{3}) + 6 ln (c)

Apply log rule:

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

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

= ln (\frac{a}{b ^{3}}) + 6 ln (c)

Apply log rule:

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

6 ln (c) = ln (c^{6})

= ln (\frac{a}{b ^{3}}) + ln (c^{6})

Apply log rule:

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

ln (\frac{a}{b ^{3}}) + ln (c^{6}) = ln (c^{6} \frac{a}{b ^{3}})

= ln (c^{6} \frac{a}{b ^{3}})

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= ln (\frac{a c ^{6}}{b ^{3}})

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What is ln(a)-3ln(b)+6ln(c) ?
The solution to ln(a)-3ln(b)+6ln(c) is ln((ac^6)/(b^3))