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

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

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

simplify

3 ln (x) - 2 ln (2 x)

ln (\frac{x}{4})

Solution steps

3 ln (x) - 2 ln (2 x)

Apply log rule:

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

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

= ln (x^{3}) - 2 ln (2 x)

Apply log rule:

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

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

= ln (x^{3}) - ln ((2 x)^{2})

Apply log rule:

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

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

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

Simplify

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

= ln (\frac{x}{4})

9.24 + lo g_{10} (\frac{0.21}{0.24})

lo g_{3} (b) + lo g_{3} (b + 9)

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

- 3 lo g_{10} (z) + 7 lo g_{10} (x) - 5 lo g_{10} (y)

lo g_{6} (x z^{3})