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

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

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

simplify

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

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

Solution steps

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

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 - 1)

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 - 1)

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 - 1)

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

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

Multiply fractions:

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

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

ln (\frac{13}{15})

lo g_{2} (\frac{4}{x + 9})

- 8 lo g_{6} (- 4 n) + 16

lo g_{10} ((5 x - 3)^{2}) + lo g_{10} ((2 x + 3)^{2}) 0

20 lo g_{5} (x) - 4 lo g_{5} (y)

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