What is 6ln(x)-1/4 ln(y)+7ln(z) ?

The solution to 6ln(x)-1/4 ln(y)+7ln(z) is ln((x^6z^7y^{3/4})/y)

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simplify 6ln(x)-1/4 ln(y)+7ln(z)

simplify

6 ln (x) - \frac{1}{4} ln (y) + 7 ln (z)

ln (\frac{x ^{6} z ^{7} y ^{\frac{3}{4}}}{y})

Solution steps

6 ln (x) - \frac{1}{4} ln (y) + 7 ln (z)

Apply log rule:

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

6 ln (x) = ln (x^{6})

= ln (x^{6}) - \frac{1}{4} ln (y) + 7 ln (z)

Apply log rule:

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

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

= ln (x^{6}) - ln (y^{\frac{1}{4}}) + 7 ln (z)

Apply log rule:

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

ln (x^{6}) - ln (y^{\frac{1}{4}}) = ln (\frac{x ^{6}}{y ^{\frac{1}{4}}})

= ln (\frac{x ^{6}}{y ^{\frac{1}{4}}}) + 7 ln (z)

Simplify

\frac{x ^{6}}{y ^{\frac{1}{4}}} : \frac{x ^{6} y ^{\frac{3}{4}}}{y}

= ln (\frac{x ^{6} y ^{\frac{3}{4}}}{y}) + 7 ln (z)

Apply log rule:

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

7 ln (z) = ln (z^{7})

= ln (\frac{x ^{6} y ^{\frac{3}{4}}}{y}) + ln (z^{7})

Apply log rule:

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

ln (\frac{x ^{6} y ^{\frac{3}{4}}}{y}) + ln (z^{7}) = ln (\frac{x ^{6} y ^{\frac{3}{4}}}{y} z^{7})

= ln (\frac{x ^{6} y ^{\frac{3}{4}}}{y} z^{7})

Simplify

\frac{x ^{6} y ^{\frac{3}{4}}}{y} z^{7} : \frac{x ^{6} z ^{7} y ^{\frac{3}{4}}}{y}

= ln (\frac{x ^{6} z ^{7} y ^{\frac{3}{4}}}{y})

lo g_{10} (\frac{100 x ^{2}}{27})

lo g_{10} (\frac{( 3 x ^{3} ) ^{4}}{( 9 x ^{8} ) ^{\frac{1}{2}}})

2 lo g_{2} (5) + lo g_{2} (3) - lo g_{2} (15)

3 lo g_{2} (A) + 5 lo g_{2} (B) - 2 lo g_{2} (C)

ln (\frac{1}{θ} e^{- \frac{x}{θ}})

What is 6ln(x)-1/4 ln(y)+7ln(z) ?
The solution to 6ln(x)-1/4 ln(y)+7ln(z) is ln((x^6z^7y^{3/4})/y)