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

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

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

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

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

3 lo g_{10} (x) + lo g_{10} (2) - lo g_{10} (y)

lo g_{4} (\frac{f}{64})

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

lo g_{2} (\frac{f}{8})

ln (x^{11} 4 - x)

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