What is 4ln(x^2)-5ln(w)-7/3 ln(z)+2ln(y^3) ?

The solution to 4ln(x^2)-5ln(w)-7/3 ln(z)+2ln(y^3) is ln((x^8y^6)/(w^5z^{7/3)})

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simplify 4ln(x^2)-5ln(w)-7/3 ln(z)+2ln(y^3)

simplify

4 ln (x^{2}) - 5 ln (w) - \frac{7}{3} ln (z) + 2 ln (y^{3})

ln (\frac{x ^{8} y ^{6}}{w ^{5} z ^{\frac{7}{3}}})

Solution steps

4 ln (x^{2}) - 5 ln (w) - \frac{7}{3} ln (z) + 2 ln (y^{3})

Apply log rule:

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

4 ln (x^{2}) = ln ((x^{2})^{4})

= ln ((x^{2})^{4}) - 5 ln (w) - \frac{7}{3} ln (z) + 2 ln (y^{3})

Apply log rule:

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

5 ln (w) = ln (w^{5})

= ln ((x^{2})^{4}) - ln (w^{5}) - \frac{7}{3} ln (z) + 2 ln (y^{3})

Apply log rule:

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

ln ((x^{2})^{4}) - ln (w^{5}) = ln (\frac{( x ^{2} ) ^{4}}{w ^{5}})

= ln (\frac{( x ^{2} ) ^{4}}{w ^{5}}) - \frac{7}{3} ln (z) + 2 ln (y^{3})

(x^{2})^{4} = x^{8}

= ln (\frac{x ^{8}}{w ^{5}}) - \frac{7}{3} ln (z) + 2 ln (y^{3})

Apply log rule:

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

\frac{7}{3} ln (z) = ln (z^{\frac{7}{3}})

= ln (\frac{x ^{8}}{w ^{5}}) - ln (z^{\frac{7}{3}}) + 2 ln (y^{3})

Apply log rule:

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

ln (\frac{x ^{8}}{w ^{5}}) - ln (z^{\frac{7}{3}}) = ln (\frac{\frac{x ^{8}}{w ^{5}}}{z ^{\frac{7}{3}}})

= ln (\frac{\frac{x ^{8}}{w ^{5}}}{z ^{\frac{7}{3}}}) + 2 ln (y^{3})

Apply the fraction rule:

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

= ln (\frac{x ^{8}}{w ^{5} z ^{\frac{7}{3}}}) + 2 ln (y^{3})

Apply log rule:

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

2 ln (y^{3}) = ln ((y^{3})^{2})

= ln (\frac{x ^{8}}{w ^{5} z ^{\frac{7}{3}}}) + ln ((y^{3})^{2})

Apply log rule:

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

ln (\frac{x ^{8}}{w ^{5} z ^{\frac{7}{3}}}) + ln ((y^{3})^{2}) = ln (\frac{x ^{8}}{w ^{5} z ^{\frac{7}{3}}} (y^{3})^{2})

= ln (\frac{x ^{8}}{w ^{5} z ^{\frac{7}{3}}} (y^{3})^{2})

Simplify

\frac{x ^{8}}{w ^{5} z ^{\frac{7}{3}}} (y^{3})^{2} : \frac{x ^{8} y ^{6}}{w ^{5} z ^{\frac{7}{3}}}

= ln (\frac{x ^{8} y ^{6}}{w ^{5} z ^{\frac{7}{3}}})

ln (x) - ln (3)

8 ln (x + 9) - 4 ln (x)

lo g_{10} (4 x + 5) - lo g_{10} (x)

lo g_{10} (\frac{2}{y})

lo g_{10} (x \cdot y)

What is 4ln(x^2)-5ln(w)-7/3 ln(z)+2ln(y^3) ?
The solution to 4ln(x^2)-5ln(w)-7/3 ln(z)+2ln(y^3) is ln((x^8y^6)/(w^5z^{7/3)})