What is 3ln(x-1)+4ln(x-6)-1/3 ln(2x-1) ?

The solution to 3ln(x-1)+4ln(x-6)-1/3 ln(2x-1) is ln(((x-1)^3(x-6)^4(2x-1)^{2/3})/(2x-1))

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simplify 3ln(x-1)+4ln(x-6)-1/3 ln(2x-1)

simplify

3 ln (x - 1) + 4 ln (x - 6) - \frac{1}{3} ln (2 x - 1)

ln (\frac{( x - 1 ) ^{3} ( x - 6 ) ^{4} ( 2 x - 1 ) ^{\frac{2}{3}}}{2 x - 1})

Solution steps

3 ln (x - 1) + 4 ln (x - 6) - \frac{1}{3} ln (2 x - 1)

Apply log rule:

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

3 ln (x - 1) = ln ((x - 1)^{3})

= ln ((x - 1)^{3}) + 4 ln (x - 6) - \frac{1}{3} ln (2 x - 1)

Apply log rule:

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

4 ln (x - 6) = ln ((x - 6)^{4})

= ln ((x - 1)^{3}) + ln ((x - 6)^{4}) - \frac{1}{3} ln (2 x - 1)

Apply log rule:

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

ln ((x - 1)^{3}) + ln ((x - 6)^{4}) = ln ((x - 6)^{4} (x - 1)^{3})

= ln ((x - 1)^{3} (x - 6)^{4}) - \frac{1}{3} ln (2 x - 1)

Apply log rule:

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

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

= ln ((x - 1)^{3} (x - 6)^{4}) - ln ((2 x - 1)^{\frac{1}{3}})

Apply log rule:

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

ln ((x - 1)^{3} (x - 6)^{4}) - ln ((2 x - 1)^{\frac{1}{3}}) = ln (\frac{( x - 6 ) ^{4} ( x - 1 ) ^{3}}{( 2 x - 1 ) ^{\frac{1}{3}}})

= ln (\frac{( x - 1 ) ^{3} ( x - 6 ) ^{4}}{( 2 x - 1 ) ^{\frac{1}{3}}})

Simplify

\frac{( x - 1 ) ^{3} ( x - 6 ) ^{4}}{( 2 x - 1 ) ^{\frac{1}{3}}} : \frac{( x - 1 ) ^{3} ( x - 6 ) ^{4} ( 2 x - 1 ) ^{\frac{2}{3}}}{2 x - 1}

= ln (\frac{( x - 1 ) ^{3} ( x - 6 ) ^{4} ( 2 x - 1 ) ^{\frac{2}{3}}}{2 x - 1})

\frac{6}{lo g _{10} ( 0.593 ) - lo g _{10} ( 0.451 )}

7 lo g_{5} (x) + 3 lo g_{5} (y^{4})

ln (x^{5})

\frac{1}{2} [3 ln (e) + ln (x) - ln (y)]

- lo g_{10} (1.8 (1 0^{- 5}))

What is 3ln(x-1)+4ln(x-6)-1/3 ln(2x-1) ?
The solution to 3ln(x-1)+4ln(x-6)-1/3 ln(2x-1) is ln(((x-1)^3(x-6)^4(2x-1)^{2/3})/(2x-1))