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

The solution to 2ln(x-1)-1/2 ln(y)-3ln(z) is ln((sqrt(y)(x-1)^2)/(yz^3))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

2 ln (x - 1) = ln ((x - 1)^{2})

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

Apply log rule:

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

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

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

Simplify

\frac{( x - 1 ) ^{2}}{y ^{\frac{1}{2}}} : \frac{y ( x - 1 ) ^{2}}{y}

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply the fraction rule:

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

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

\frac{2}{3} lo g_{a} (- 2 lo g_{b} (- \frac{1}{5} lo g_{c} (x)))

ln (2) - ln (10)

lo g_{10} (\frac{5}{x})

lo g_{2} ((x + y) \cdot z)

ln (x^{2} - 9) - ln (x + 3)

What is 2ln(x-1)-1/2 ln(y)-3ln(z) ?
The solution to 2ln(x-1)-1/2 ln(y)-3ln(z) is ln((sqrt(y)(x-1)^2)/(yz^3))