What is 3/4 ln(x)+ln(sqrt(x^2+1))-7ln(4x+5) ?

The solution to 3/4 ln(x)+ln(sqrt(x^2+1))-7ln(4x+5) is ln((x^{3/4}sqrt(x^2+1))/((4x+5)^7))

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simplify 3/4 ln(x)+ln(sqrt(x^2+1))-7ln(4x+5)

simplify

\frac{3}{4} ln (x) + ln (x^{2} + 1) - 7 ln (4 x + 5)

ln (\frac{x ^{\frac{3}{4}} x ^{2} + 1}{( 4 x + 5 ) ^{7}})

Solution steps

\frac{3}{4} ln (x) + ln (x^{2} + 1) - 7 ln (4 x + 5)

Apply log rule:

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

\frac{3}{4} ln (x) = ln (x^{\frac{3}{4}})

= ln (x^{\frac{3}{4}}) + ln (x^{2} + 1) - 7 ln (4 x + 5)

Apply log rule:

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

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

= ln (x^{\frac{3}{4}} x^{2} + 1) - 7 ln (4 x + 5)

Apply log rule:

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

7 ln (4 x + 5) = ln ((4 x + 5)^{7})

= ln (x^{\frac{3}{4}} x^{2} + 1) - ln ((4 x + 5)^{7})

Apply log rule:

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

ln (x^{\frac{3}{4}} x^{2} + 1) - ln ((4 x + 5)^{7}) = ln (\frac{x ^{\frac{3}{4}} x ^{2} + 1}{( 4 x + 5 ) ^{7}})

= ln (\frac{x ^{\frac{3}{4}} x ^{2} + 1}{( 4 x + 5 ) ^{7}})

α ln (\frac{α q}{α + 1}) + ln (\frac{q}{( α + 1 ) ^{2}})

lo g_{4} (4 x) + 2 lo g_{4} (4)

lo g_{3} (3 x^{9} x^{27} x^{729})

lo g_{3} (x^{2} y^{5})

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What is 3/4 ln(x)+ln(sqrt(x^2+1))-7ln(4x+5) ?
The solution to 3/4 ln(x)+ln(sqrt(x^2+1))-7ln(4x+5) is ln((x^{3/4}sqrt(x^2+1))/((4x+5)^7))