What is 1/2 ln(9x^2)+2ln(x)-3ln(2x) ?

The solution to 1/2 ln(9x^2)+2ln(x)-3ln(2x) is ln(3/8)

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simplify 1/2 ln(9x^2)+2ln(x)-3ln(2x)

simplify

\frac{1}{2} ln (9 x^{2}) + 2 ln (x) - 3 ln (2 x)

ln (\frac{3}{8})

Decimal Notation

- 0.98082 \dots

Solution steps

\frac{1}{2} ln (9 x^{2}) + 2 ln (x) - 3 ln (2 x)

Apply log rule:

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

\frac{1}{2} ln (9 x^{2}) = ln ((9 x^{2})^{\frac{1}{2}})

= ln ((9 x^{2})^{\frac{1}{2}}) + 2 ln (x) - 3 ln (2 x)

Apply log rule:

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

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

= ln ((9 x^{2})^{\frac{1}{2}}) + ln (x^{2}) - 3 ln (2 x)

Apply log rule:

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

ln ((9 x^{2})^{\frac{1}{2}}) + ln (x^{2}) = ln (x^{2} (9 x^{2})^{\frac{1}{2}})

= ln (x^{2} (9 x^{2})^{\frac{1}{2}}) - 3 ln (2 x)

Simplify

(9 x^{2})^{\frac{1}{2}} x^{2} : 3 x^{3}

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

\frac{3 x ^{3}}{( 2 x ) ^{3}} = \frac{3}{8}

= ln (\frac{3}{8})

- 2 ln ∣ ln (x) ∣ - ln (x)

10 \cdot ln (0.5 \cdot X)

6 lo g_{9} (x) - 3 lo g_{9} (y)

- lo g_{10} (8.7 \cdot 1 0^{- 9})

\frac{1}{2} lo g_{10} (x) + lo g_{10} (y)

What is 1/2 ln(9x^2)+2ln(x)-3ln(2x) ?
The solution to 1/2 ln(9x^2)+2ln(x)-3ln(2x) is ln(3/8)