What is ln(x)-3ln(y)-ln(4) ?

The solution to ln(x)-3ln(y)-ln(4) is ln(x/(4y^3))

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simplify ln(x)-3ln(y)-ln(4)

simplify

ln (x) - 3 ln (y) - ln (4)

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

Solution steps

ln (x) - 3 ln (y) - ln (4)

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply the fraction rule:

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

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

ln (6 + x^{2}) - ln (3 + x)

5.35 \cdot ln (\frac{693}{298}) + 2.4 \cdot 1 0^{- 3} \cdot (693 - 298)

lo g_{10} (4 x^{4} \cdot (1 - x)^{4} \cdot (1 - 2 x)^{2})

ln ∣ x ∣ - ln (x^{2} + 1)

5 lo g_{8} (7 z + 1) + \frac{1}{2} lo g_{8} (z + 7)