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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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