What is ln((x^8sqrt(p^7q^9))/(e^9)) ?

The solution to ln((x^8sqrt(p^7q^9))/(e^9)) is 8ln(x)+ln(sqrt(p)sqrt(q)sqrt(p^6)sqrt(q^8))-9

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simplify ln((x^8sqrt(p^7q^9))/(e^9))

simplify

ln (\frac{x ^{8} p ^{7} q ^{9}}{e ^{9}})

8 ln (x) + ln (p q p^{6} q^{8}) - 9

Solution steps

ln (\frac{x ^{8} p ^{7} q ^{9}}{e ^{9}})

Apply log rule:

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

ln (\frac{x ^{8} p ^{7} q ^{9}}{e ^{9}}) = ln (x^{8} p^{7} q^{9}) - ln (e^{9})

= ln (x^{8} p^{7} q^{9}) - ln (e^{9})

Apply log rule:

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

ln (e^{9}) = 9 ln (e)

= ln (x^{8} p^{7} q^{9}) - 9 ln (e)

Apply log rule:

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

ln (x^{8} p^{7} q^{9}) = ln (x^{8}) + ln (p^{7} q^{9})

= ln (x^{8}) + ln (p^{7} q^{9}) - 9 ln (e)

Apply log rule:

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

ln (x^{8}) = 8 ln (x)

= 8 ln (x) + ln (p^{7} q^{9}) - 9 ln (e)

ln (p^{7} q^{9}) = ln (p q p^{6} q^{8})

9 ln (e) = 9

= 8 ln (x) + ln (p q p^{6} q^{8}) - 9

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- lo g_{10} (a) + 2 lo g_{10} (b) - 1

lo g_{10} (\frac{x ^{2}}{( x - 5 ) ^{2}})

ln (\frac{( 1 + x )}{( 1 - x )})

\frac{ln ( \frac{41}{16} )}{4 ln ( 2 )}

What is ln((x^8sqrt(p^7q^9))/(e^9)) ?
The solution to ln((x^8sqrt(p^7q^9))/(e^9)) is 8ln(x)+ln(sqrt(p)sqrt(q)sqrt(p^6)sqrt(q^8))-9