What is 2*ln(x)+1/3*ln(x-2)-5*ln(2x+3) ?

The solution to 2*ln(x)+1/3*ln(x-2)-5*ln(2x+3) is ln((x^2(x-2)^{1/3})/((2x+3)^5))

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simplify 2ln(x)+1/3ln(x-2)-5*ln(2x+3)

simplify

2 \cdot ln (x) + \frac{1}{3} \cdot ln (x - 2) - 5 \cdot ln (2 x + 3)

ln (\frac{x ^{2} ( x - 2 ) ^{\frac{1}{3}}}{( 2 x + 3 ) ^{5}})

Solution steps

2 ln (x) + \frac{1}{3} ln (x - 2) - 5 ln (2 x + 3)

Apply log rule:

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

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

= ln (x^{2}) + \frac{1}{3} ln (x - 2) - 5 ln (2 x + 3)

Apply log rule:

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

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

= ln (x^{2}) + ln ((x - 2)^{\frac{1}{3}}) - 5 ln (2 x + 3)

Apply log rule:

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

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

= ln (x^{2} (x - 2)^{\frac{1}{3}}) - 5 ln (2 x + 3)

Apply log rule:

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

5 ln (2 x + 3) = ln ((2 x + 3)^{5})

= ln (x^{2} (x - 2)^{\frac{1}{3}}) - ln ((2 x + 3)^{5})

Apply log rule:

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

ln (x^{2} (x - 2)^{\frac{1}{3}}) - ln ((2 x + 3)^{5}) = ln (\frac{x ^{2} ( x - 2 ) ^{\frac{1}{3}}}{( 2 x + 3 ) ^{5}})

= ln (\frac{x ^{2} ( x - 2 ) ^{\frac{1}{3}}}{( 2 x + 3 ) ^{5}})

ln (3) - ln (3 - x)

27 lo g_{10} (x) + lo g_{10} (y) - 12 lo g_{10} (z)

ln (0.42) + ln (0.37) + ln (0.64) + ln (0.55)

ln (3) - 5 ln (x)

ln (r) - ln (a)

What is 2*ln(x)+1/3*ln(x-2)-5*ln(2x+3) ?
The solution to 2*ln(x)+1/3*ln(x-2)-5*ln(2x+3) is ln((x^2(x-2)^{1/3})/((2x+3)^5))