What is 2ln(3x)+5ln(x-7)-ln(x^2-1) ?

The solution to 2ln(3x)+5ln(x-7)-ln(x^2-1) is ln((9x^2(x-7)^5)/(x^2-1))

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simplify 2ln(3x)+5ln(x-7)-ln(x^2-1)

simplify

2 ln (3 x) + 5 ln (x - 7) - ln (x^{2} - 1)

ln (\frac{9 x ^{2} ( x - 7 ) ^{5}}{x ^{2} - 1})

Solution steps

2 ln (3 x) + 5 ln (x - 7) - ln (x^{2} - 1)

Apply log rule:

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

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

= ln ((3 x)^{2}) + 5 ln (x - 7) - ln (x^{2} - 1)

Apply log rule:

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

5 ln (x - 7) = ln ((x - 7)^{5})

= ln ((3 x)^{2}) + ln ((x - 7)^{5}) - ln (x^{2} - 1)

Apply log rule:

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

ln ((3 x)^{2}) + ln ((x - 7)^{5}) = ln ((3 x)^{2} (x - 7)^{5})

= ln ((3 x)^{2} (x - 7)^{5}) - ln (x^{2} - 1)

Simplify

(3 x)^{2} (x - 7)^{5} : 9 x^{2} (x - 7)^{5}

= ln (9 x^{2} (x - 7)^{5}) - ln (x^{2} - 1)

Apply log rule:

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

ln (9 x^{2} (x - 7)^{5}) - ln (x^{2} - 1) = ln (\frac{9 x ^{2} ( x - 7 ) ^{5}}{x ^{2} - 1})

= ln (\frac{9 x ^{2} ( x - 7 ) ^{5}}{x ^{2} - 1})

ln (\frac{( 2 t + 1 ) ^{3}}{( 3 t - 1 ) ^{4}})

ln (\frac{1}{a} e^{- \frac{x}{a}})

8 lo g_{e} (2 x) - lo g_{e} (2)

lo g_{2} ((x^{3} + 2) (x - 4)^{3})

lo g_{10} (5 \cdot 1 0^{- 3})

What is 2ln(3x)+5ln(x-7)-ln(x^2-1) ?
The solution to 2ln(3x)+5ln(x-7)-ln(x^2-1) is ln((9x^2(x-7)^5)/(x^2-1))