What is ln(x)-3ln(z)+12ln(y) ?

The solution to ln(x)-3ln(z)+12ln(y) is ln((xy^{12})/(z^3))

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simplify ln(x)-3ln(z)+12ln(y)

simplify

ln (x) - 3 ln (z) + 12 ln (y)

ln (\frac{x y ^{12}}{z ^{3}})

Solution steps

ln (x) - 3 ln (z) + 12 ln (y)

Apply log rule:

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

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

= ln (x) - ln (z^{3}) + 12 ln (y)

Apply log rule:

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

ln (x) - ln (z^{3}) = ln (\frac{x}{z ^{3}})

= ln (\frac{x}{z ^{3}}) + 12 ln (y)

Apply log rule:

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

12 ln (y) = ln (y^{12})

= ln (\frac{x}{z ^{3}}) + ln (y^{12})

Apply log rule:

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

ln (\frac{x}{z ^{3}}) + ln (y^{12}) = ln (\frac{x}{z ^{3}} y^{12})

= ln (\frac{x}{z ^{3}} y^{12})

Multiply fractions:

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

= ln (\frac{x y ^{12}}{z ^{3}})

2 ln (\frac{5 e}{3})

4 ln (x y) + ln (y) - ln (x^{4} y)

2 ln (5) + 3 ln (4)

ln (\frac{y}{c + x})

lo g_{2} (\frac{4 π ^{2}}{30})

What is ln(x)-3ln(z)+12ln(y) ?
The solution to ln(x)-3ln(z)+12ln(y) is ln((xy^{12})/(z^3))