What is ln(((x^5*z^5))/(y^{11)*w^{15}}) ?

The solution to ln(((x^5*z^5))/(y^{11)*w^{15}}) is 5ln(x)+5ln(z)-11ln(y)-15ln(w)

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simplify ln(((x^5z^5))/(y^{11)w^{15}})

simplify

ln (\frac{( x ^{5} \cdot z ^{5} )}{y ^{11} \cdot w ^{15}})

5 ln (x) + 5 ln (z) - 11 ln (y) - 15 ln (w)

Solution steps

ln (\frac{( x ^{5} z ^{5} )}{y ^{11} w ^{15}})

Apply log rule:

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

ln (\frac{( x ^{5} z ^{5} )}{y ^{11} w ^{15}}) = ln ((x^{5} z^{5})) - ln (y^{11} w^{15})

= ln (x^{5} z^{5}) - ln (y^{11} w^{15})

Apply log rule:

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

ln (x^{5} z^{5}) = ln (x^{5}) + ln (z^{5}), ln (y^{11} w^{15}) = ln (y^{11}) + ln (w^{15})

= ln (x^{5}) + ln (z^{5}) - (ln (y^{11}) + ln (w^{15}))

Apply log rule:

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

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

= 5 ln (x) + ln (z^{5}) - (ln (y^{11}) + ln (w^{15}))

Apply log rule:

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

ln (z^{5}) = 5 ln (z)

= 5 ln (x) + 5 ln (z) - (ln (y^{11}) + ln (w^{15}))

Apply log rule:

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

ln (y^{11}) = 11 ln (y)

= 5 ln (x) + 5 ln (z) - (11 ln (y) + ln (w^{15}))

Apply log rule:

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

ln (w^{15}) = 15 ln (w)

= 5 ln (x) + 5 ln (z) - (11 ln (y) + 15 ln (w))

- (11 ln (y) + 15 ln (w)) : - 11 ln (y) - 15 ln (w)

= 5 ln (x) + 5 ln (z) - 11 ln (y) - 15 ln (w)

\frac{5 ln ( 7 )}{3 ln ( 10 ) - ln ( 7 )}

4.745 + lo g_{10} (\frac{0.146}{0.105})

3 lo g_{10} (x) + 4 lo g_{10} (x - 1)

2 [ln (x) - ln (x + 1) + ln (x - 1)]

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

What is ln(((x^5*z^5))/(y^{11)*w^{15}}) ?
The solution to ln(((x^5*z^5))/(y^{11)*w^{15}}) is 5ln(x)+5ln(z)-11ln(y)-15ln(w)