What is 2ln(w)+3ln(w)-1/2 ln(x)-5ln(y) ?

The solution to 2ln(w)+3ln(w)-1/2 ln(x)-5ln(y) is ln((w^5sqrt(x))/(xy^5))

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simplify 2ln(w)+3ln(w)-1/2 ln(x)-5ln(y)

simplify

2 ln (w) + 3 ln (w) - \frac{1}{2} ln (x) - 5 ln (y)

ln (\frac{w ^{5} x}{x y ^{5}})

Solution steps

2 ln (w) + 3 ln (w) - \frac{1}{2} ln (x) - 5 ln (y)

Apply log rule:

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

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

= 2 ln (w) + ln (w^{3}) - \frac{1}{2} ln (x) - 5 ln (y)

Apply log rule:

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

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

= 2 ln (w) + ln (w^{3}) - ln (x^{\frac{1}{2}}) - 5 ln (y)

Apply log rule:

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

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

= 2 ln (w) + ln (\frac{w ^{3}}{x ^{\frac{1}{2}}}) - 5 ln (y)

Simplify

\frac{w ^{3}}{x ^{\frac{1}{2}}} : \frac{w ^{3} x}{x}

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

Apply log rule:

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

2 ln (w) = ln (w^{2})

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

Apply log rule:

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

ln (w^{2}) + ln (\frac{w ^{3} x}{x}) = ln (\frac{w ^{3} x}{x} w^{2})

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

Simplify

w^{2} \frac{w ^{3} x}{x} : \frac{w ^{5} x}{x}

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (\frac{\frac{w ^{5} x}{x}}{y ^{5}})

Apply the fraction rule:

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

= ln (\frac{w ^{5} x}{x y ^{5}})

5 lo g_{3} (u) + 6 lo g_{3} (v)

8 ln x + 4 - 4 + 2 ln x + 4 + 1

lo g_{10} (0.5 x)

lo g_{3} (243 (x - 1)^{4})

2 lo g_{b} (z) - lo g_{b} (y)

What is 2ln(w)+3ln(w)-1/2 ln(x)-5ln(y) ?
The solution to 2ln(w)+3ln(w)-1/2 ln(x)-5ln(y) is ln((w^5sqrt(x))/(xy^5))