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

The solution to 2/3 ln(x-2)-2ln(x+1)+ln(2x) is ln((2x(x-2)^{2/3})/((x+1)^2))

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

2 ln (x + 1) = ln ((x + 1)^{2})

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

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

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

5 lo g_{10} (x) + lo g_{10} (7)

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

- lo g_{10} (4.907 \cdot 1 0^{- 3})

ln (\frac{a}{b + c})

- lo g_{10} (1.6 \cdot 1 0^{- 11})

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