What is ln(a/b)-2ln(a^3)+4ln(b) ?

The solution to ln(a/b)-2ln(a^3)+4ln(b) is ln((b^3)/(a^5))

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simplify ln(a/b)-2ln(a^3)+4ln(b)

simplify

ln (\frac{a}{b}) - 2 ln (a^{3}) + 4 ln (b)

ln (\frac{b ^{3}}{a ^{5}})

Solution steps

ln (\frac{a}{b}) - 2 ln (a^{3}) + 4 ln (b)

Apply log rule:

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

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

= ln (\frac{a}{b}) - ln ((a^{3})^{2}) + 4 ln (b)

Apply log rule:

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

ln (\frac{a}{b}) - ln ((a^{3})^{2}) = ln (\frac{\frac{a}{b}}{( a ^{3} ) ^{2}})

= ln (\frac{\frac{a}{b}}{( a ^{3} ) ^{2}}) + 4 ln (b)

Simplify

\frac{\frac{a}{b}}{( a ^{3} ) ^{2}} : \frac{1}{a ^{5} b}

= ln (\frac{1}{a ^{5} b}) + 4 ln (b)

Apply log rule:

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

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

= ln (\frac{1}{a ^{5} b}) + ln (b^{4})

Apply log rule:

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

ln (\frac{1}{a ^{5} b}) + ln (b^{4}) = ln (b^{4} \frac{1}{a ^{5} b})

= ln (b^{4} \frac{1}{a ^{5} b})

Simplify

\frac{1}{a ^{5} b} b^{4} : \frac{b ^{3}}{a ^{5}}

= ln (\frac{b ^{3}}{a ^{5}})

\frac{π}{8} ln (2)

lo g_{2} (5) + lo g_{2} (12) - lo g_{2} (3.75)

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

lo g_{10} (w) + lo g_{10} (v) - 3 lo g_{10} (z)

ln (\frac{205}{200}) + (\frac{1}{2} \cdot 0.0 2^{2} - 0.05) \cdot 0.25

What is ln(a/b)-2ln(a^3)+4ln(b) ?
The solution to ln(a/b)-2ln(a^3)+4ln(b) is ln((b^3)/(a^5))