What is ln(2x)+3(ln(x)-ln(y)) ?

The solution to ln(2x)+3(ln(x)-ln(y)) is ln((2x^4)/(y^3))

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simplify ln(2x)+3(ln(x)-ln(y))

simplify

ln (2 x) + 3 (ln (x) - ln (y))

ln (\frac{2 x ^{4}}{y ^{3}})

Solution steps

ln (2 x) + 3 (ln (x) - ln (y))

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Simplify

2 x (\frac{x}{y})^{3} : \frac{2 x ^{4}}{y ^{3}}

= ln (\frac{2 x ^{4}}{y ^{3}})

\frac{1}{2} ln (x + 3) + \frac{1}{2} ln (x + 2)

lo g_{2} (\frac{1}{x}) - lo g_{2} (\frac{1}{x ^{2}})

lo g_{10} (\frac{y ^{3}}{2 x})

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

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

What is ln(2x)+3(ln(x)-ln(y)) ?
The solution to ln(2x)+3(ln(x)-ln(y)) is ln((2x^4)/(y^3))