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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

ln (x) - ln (y) = ln (\frac{x}{y})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

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

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

4 lo g_{2} (11) - 6 lo g_{2} (6)

ln (\frac{( 3 ^{x} 6 ^{2 x} )}{2 ^{3 x}})

ln (13 x) - ln (13)

5 ln (x) + ln (y)

lo g_{9} (4) + 3 lo g_{9} (x - 1)

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