What is 3ln(x)+1/2 ln(y)-4ln(z) ?

The solution to 3ln(x)+1/2 ln(y)-4ln(z) is ln((x^3y^{1/2})/(z^4))

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simplify 3ln(x)+1/2 ln(y)-4ln(z)

simplify

3 ln (x) + \frac{1}{2} ln (y) - 4 ln (z)

ln (\frac{x ^{3} y ^{\frac{1}{2}}}{z ^{4}})

Solution steps

3 ln (x) + \frac{1}{2} ln (y) - 4 ln (z)

Apply log rule:

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

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

= ln (x^{3}) + \frac{1}{2} ln (y) - 4 ln (z)

Apply log rule:

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

\frac{1}{2} ln (y) = ln (y^{\frac{1}{2}})

= ln (x^{3}) + ln (y^{\frac{1}{2}}) - 4 ln (z)

Apply log rule:

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

ln (x^{3}) + ln (y^{\frac{1}{2}}) = ln (x^{3} y^{\frac{1}{2}})

= ln (x^{3} y^{\frac{1}{2}}) - 4 ln (z)

Apply log rule:

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

4 ln (z) = ln (z^{4})

= ln (x^{3} y^{\frac{1}{2}}) - ln (z^{4})

Apply log rule:

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

ln (x^{3} y^{\frac{1}{2}}) - ln (z^{4}) = ln (\frac{x ^{3} y ^{\frac{1}{2}}}{z ^{4}})

= ln (\frac{x ^{3} y ^{\frac{1}{2}}}{z ^{4}})

lo g_{10} (x + 2) + lo g_{10} (x - 1)

lo g_{b} (\frac{7}{6})

ln (\frac{x ^{2} x ^{2} - 1}{e ^{x}})

\frac{1}{3} lo g_{b} (x) + 4 lo g_{b} (y) - 2 lo g_{b} (x)

2 lo g_{10} (2) + 3 lo g_{10} (3) - 2 lo g_{10} (6)

What is 3ln(x)+1/2 ln(y)-4ln(z) ?
The solution to 3ln(x)+1/2 ln(y)-4ln(z) is ln((x^3y^{1/2})/(z^4))