What is ln((6x^3e^{2x-7})/(sqrt(x^2+9))) ?

The solution to ln((6x^3e^{2x-7})/(sqrt(x^2+9))) is ln(6)+3ln(x)+2x-7-ln(sqrt(x^2+9))

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simplify ln((6x^3e^{2x-7})/(sqrt(x^2+9)))

simplify

ln (\frac{6 x ^{3} e ^{2 x - 7}}{x ^{2} + 9})

ln (6) + 3 ln (x) + 2 x - 7 - ln (x^{2} + 9)

Solution steps

ln (\frac{6 x ^{3} e ^{2 x - 7}}{x ^{2} + 9})

Apply log rule:

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

ln (\frac{6 x ^{3} e ^{2 x - 7}}{x ^{2} + 9}) = ln (6 x^{3} e^{2 x - 7}) - ln (x^{2} + 9)

= ln (6 e^{2 x - 7} x^{3}) - ln (x^{2} + 9)

Apply log rule:

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

ln (6 x^{3} e^{2 x - 7}) = ln (6) + ln (x^{3}) + ln (e^{2 x - 7})

= ln (6) + ln (x^{3}) + ln (e^{2 x - 7}) - ln (x^{2} + 9)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= ln (6) + 3 ln (x) + ln (e^{2 x - 7}) - ln (x^{2} + 9)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (e^{2 x - 7}) = (2 x - 7) ln (e)

= ln (6) + 3 ln (x) + (2 x - 7) ln (e) - ln (x^{2} + 9)

(2 x - 7) ln (e) = 2 x - 7

= ln (6) + 3 ln (x) + 2 x - 7 - ln (x^{2} + 9)

∣ (7 - 3 x) ∣ < 2

\frac{x ^{2} - 4}{x ^{2} - 9} < 0

- 12 = (f - 6)

\frac{5}{2} + 3

3 x^{2} - 16 - 12

What is ln((6x^3e^{2x-7})/(sqrt(x^2+9))) ?
The solution to ln((6x^3e^{2x-7})/(sqrt(x^2+9))) is ln(6)+3ln(x)+2x-7-ln(sqrt(x^2+9))